Wolff potentials and measure data vectorial problems with Orlicz growth

Abstract

We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for p-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth.

1 Introduction

A broad and profuse branch of the theory of nonlinear partial differential systems stems from seminal ideas by Ural’tseva [85] and Uhlenbeck [84]. There is a solid stream of recent studies on regularity of solutions to general growth elliptic systems [4, 25, 33, 38, 67, 72, 81] and related minimizers of vectorial functionals [7, 15, 30, 32, 35, 36, 47, 73]. Our aim is to contribute to the field by providing pointwise estimates for very weak solutions to measure data problems in the terms of potentials of relevant Orlicz growth. As a consequence, we infer sharp description of fine properties of solutions being the exact analogues of the ones available in the classical linear potential theory or in the case of p-Laplace systems [65].

Let us stress that weak solutions do not have to exist for arbitrary measure datum. Thus we employ a notion of very weak solutions obtained by an approximation studied since [9]. Despite they can be unbounded, they can be controlled by a certain potential. There are known deep classical results for scalar problems [57, 58] settling the nonlinear potential theory for solutions to $-\,\Delta _p u=-\,\textrm{div}(|Du|^{p-2}Du)=\mu$, $1<p<\infty$, followed by their variable exponent version [68] and recently by the Orlicz one [22, 70], as well as a counterpart proven for systems involving p-Laplace operator [27, 65]. We study the nonstandard growth version of pointwise estimates involving suitably generalized potential of the Wolff type and infer their regularity consequences.

In fact, we investigate very weak solutions ${{\varvec{\textsf {u}}}}:\Omega \rightarrow {{\mathbb {R}}^{m}}$ to measure data elliptic systems involving nonlinear operators

$$\begin{aligned} -{ {{\varvec{{\mathsf{\textrm{div}}}}}}}\left( {a(x)}\frac{g(|{D{\varvec{\textsf {u}}}}|)}{|{D{\varvec{\textsf {u}}}}|} {D{\varvec{\textsf {u}}}}\right) ={{\varvec{{\mathsf{\mu }}}}}\quad \text {in}\quad \Omega , \end{aligned}$$

(1)

where $a\in C(\Omega )$ is bounded and separated from zero, $g=G'$, ${{\varvec{{\mathsf{\mu }}}}}\in {\mathcal {M}}(\Omega ,{{\mathbb {R}}^{m}})$ is a bounded measure, whereas ${{\varvec{{\mathsf{\textrm{div}}}}}}$ stands for the ${{\mathbb {R}}^{m}}$-valued divergence operator. Here we admit G to be a Young function satisfying both $\Delta _2$- and $\nabla _2$-conditions, which follows from the condition

$$\begin{aligned} 2 \le i_G=\inf _{t>0}\frac{tg(t)}{G(t)}\le \sup _{t>0}\frac{tg(t)}{G(t)}=s_G<\infty . \end{aligned}$$

(2)

See Assumption (A-vect) in Sect. 2 for all details. We stress that we impose the typical assumption of quasi-diagonal structure of ${{\mathcal {A}}}$ naturally covering the case of (possibly weighted) p-Laplacian when $G_p(s)=s^{p},$ for every $p\ge 2$, together with operators governed by the Zygmund-type functions $G_{p,\alpha }(s)=s^p\log ^\alpha (1+s)$, $p\ge 2,\,\alpha \in {{\mathbb {R}}}$, as well as their multiplications and compositions with various parameters. In turn, we generalize the corresponding results of [27, 65] to embrace also p-Laplace systems with continuous coefficients, and – on the other hand – cover the natural scope of operators with similar structure and Orlicz growth. Let us stress that the operator we consider is not assumed to enjoy homogeneity of a form ${{\mathcal {A}}}(x,k\xi )=|k|^{p-2}k{{\mathcal {A}}}(x,\xi )$. Consequently, our class of solutions is not invariant with respect to scalar multiplication.

Obtaining sharp regularity results for solutions to nonlinear systems is particularly challenging. In the scalar case one can infer continuity or Hölder continuity of the solution to $-\textrm{div}{{\mathcal {A}}}(x,Du)=\mu$ when the dependence of the operator on the spacial variable is merely bounded and measurable and the growth of ${{\mathcal {A}}}$ with respect to the second variable is governed by an arbitrary doubling Young function (cf. [22]). The same is not possible for systems even with far less complicated growth and null datum, cf. [29, 55, 82] and [75, Sect. 3]. To justify why we restrict our attention to operators having a specific form as in (1), let us point out that the typical assumption of a so-called Uhlenbeck structure is imposed in order to control energy of solutions. The continuity of the coefficients is a minimal assumption to get continuity of the solution in the view of counterexample of [29].

The studies on the potential theory to measure data problems dates back to [52, 74]. We refer to [62] for an overview of the nonlinear potential theory, to [54, 57, 58, 69] for cornerstones of the field, and [1, 53] for well-present background of the p-growth case. In the scalar case, in their seminal works [57, 58], Kilpeläinen and Malý provided optimal Wolff potential estimates for p-superharmonic functions u generating a nonnegative measure $\mu$ from above and below

$$\begin{aligned} \tfrac{1}{c}{{\mathcal {W}}}_p^{\mu }(x_0,R)\le u(x_0)\le c\Big (\inf _{B_R(x_0)}u+{{\mathcal {W}}}_p^{\mu }(x_0,R)\Big )\quad \text {for some }\ c=c(n,p) \end{aligned}$$

(3)

with the so-called Wolff potential

$$\begin{aligned} {{\mathcal {W}}}_p^{\mu }(x_0,R)=\int _0^R \left( r^{p-n}{\mu (B_r(x_0))}\right) ^{\frac{1}{p-1}}\,\frac{dr}{r}= \int _0^R \left( \frac{\mu (B_r(x_0))}{r^{n-1}}\right) ^{\frac{1}{p-1}}\,dr, \end{aligned}$$

see also [60, 83]. In the linear case ($p=2$) the estimates of (3) become the classical Riesz potential bounds. The precise Orlicz counterpart of this result with nonnegative measure $\mu$ is proven with the nonstandard growth potential

$$\begin{aligned} {{\mathcal {W}}}^{\mu }_G(x_0,R)=\int _0^R g^{-1}\left( \frac{\mu (B_r(x_0))}{r^{n-1}}\right) \,dr, \end{aligned}$$

see [22, 70]. To our best knowledge, the only reference one can find on the related results for systems are [27, 65] that involves problems with the p-Laplace operator. The estimate related to (3) provided therein establishes the upper bound only. Note however that no lower bound can be available in the vectorial case. Indeed, it origins in the lack of the possibility of proving maximum principle. Our method of proof relies on the ideas of [65]. We employ a properly adapted Orlicz version of ${{\mathcal {A}}}$-harmonic approximation relevant for measure data problems (Theorem 4.1 in Sect. 4) and careful estimates on concentric balls. The proof of Theorem 4.1 is based on the arguments that essentially apply the Uhlebeck structure of the system and the lower growth restriction (2). By the very nature of the result one cannot hope for ${{\mathcal {A}}}$-harmonic approximation of functions that do not have at least Sobolev regularity. In order to ensure that a distributional solution to a measure data problem belongs at least to $W^{1,1}_{loc}$, the operator need to be far from 1-Laplacian. Note that already in the case of scalar p-Laplace equation, the value $p=2-\frac{1}{n}$ is the integrability threshold for a fundamental solution. Indeed, for $p\ne n$ a function $u(x)=c(|x|^\frac{p-n}{n-1}-1)$, being a solution to $-\Delta _p u=\delta _0$ on a ball, has locally integrable distributional gradient only for $p>2-\frac{1}{n}$. Similarly, one can show the natural Orlicz counterpart of this threshold. If this condition is violated, a fundamental part of the proof of Theorem 4.1, i.e. the estimate on summability of ${\varvec{\textsf {u}}}$ and $D{\varvec{\textsf {u}}}$, is false. Note that Remark 4.2 explains that the proof of Theorem 4.1 works for functions G having $i_G$ slightly below 2. Other available proofs of summability estimates for solutions to similar problems with Orlicz growth also do not cover the natural range reflecting $p>2-\frac{1}{n}$, see [4]. We stress that rest of the proofs, including those of pointwise Wolff potential estimate (Theorem 2.1), continuity criterium (Theorem 2.3) and Hölder continuity criterium (Theorem 2.9), do not make use of other growth assumptions than $G\in \Delta _2\cap \nabla _2$. Nonetheless, all of them substantially rely on measure data ${{\mathcal {A}}}$-harmonic approximation result.

Let us also mention that a lot of attention is attracted by potential estimates on gradients of solutions, generalized harmonic approximation, and their application in the theory of partial regularity [5, 6, 10,11,12,13, 16, 17, 19, 34, 37, 41,42,43,44, 62, 64, 71, 76] which is an open path from now on.

The paper is organized as follows. Our assumptions, main results and their regularity consequences are presented in Sect. 2. Sect. 3 is devoted to notation and information on the setting. In particular see Sect. 3.5 for the precise definition of the notion of very weak solutions we employ. Sect. 4 provides the most important tool of paper—a measure data ${{\mathcal {A}}}$-harmonic approximation. Sect. 5 contains the proofs of comparison estimates, the sufficient condition for ${\varvec{\textsf {u}}}$ to be in VMO of Proposition 2.2, the potential estimates of Theorem 2.1, the continuity criterion of Theorem 2.3 and the Hölder continuity criterion of Theorem 2.9.

2 Main result and its consequences

2.1 The statement of the problem

Let us present an essential notation and details of the measure data problem we study.

Essential notation. By ‘$\cdot$’ we denote the scalar product of two vectors, i.e. for ${{{\varvec{{\mathsf{\xi }}}}}}=(\xi _1,\dots ,\xi _m)\in {{\mathbb {R}}^{m}}$ and ${{\varvec{{\mathsf{\eta }}}}}= (\eta _1,\dots ,\eta _m)\in {{\mathbb {R}}^{m}}$ we have ${{{\varvec{{\mathsf{\xi }}}}}}\cdot {{{\varvec{{\mathsf{\eta }}}}}} = \sum _{i=1}^m \xi _i \eta _i$; by ‘ : ’ – the Frobenius product of the second-order tensors, i.e. for ${\xi }=[\xi _{j}^\alpha ]_{j=1,\dots ,n,\, \alpha =1,\dots ,m}$ and $\eta =[\eta _{j}^\alpha ]_{j=1,\dots ,n,\, \alpha =1,\dots ,m}$ we have

$$\begin{aligned} {\xi }: {\eta } =\sum _{\alpha =1}^m \sum _{j=1}^n \xi _{j}^\alpha \eta _{j}^\alpha . \end{aligned}$$

By ‘$\otimes$’ we denote the tensor product of two vectors, i.e for ${{{{\varvec{{\mathsf{\xi }}}}}}}=(\xi _1,\dots ,\xi _k)\in {{{\mathbb {R}}}^k}$ and ${{{{\varvec{{\mathsf{\eta }}}}}}}= (\eta _1,\dots ,\eta _\ell )\in {{\mathbb {R}}}^\ell$, we have ${{\varvec{{\mathsf{\xi }}}}}\otimes {{\varvec{{\mathsf{\eta }}}}}:=[\xi _i\eta _j]_{i=1,\dots ,k,\,j=1,\dots ,\ell },$ that is

$$\begin{aligned} {{{\varvec{{\mathsf{\xi }}}}}}\otimes {{{\varvec{{\mathsf{\eta }}}}}}:=\begin{pmatrix} \xi _{1}\eta _{1} &{} \xi _{1}\eta _{2} &{} \cdots &{} \xi _{1}\eta _{\ell } \\ \xi _{2}\eta _{1} &{} \xi _{2}\eta _{2} &{} \cdots &{} \xi _{2}\eta _{\ell } \\ \vdots &{} \vdots &{} &{} \vdots \\ \xi _{k}\eta _{1} &{} \xi _{k}\eta _{2} &{} \cdots &{} \xi _{k}\eta _{\ell } \end{pmatrix}\in {{\mathbb {R}}}^{k\times \ell }. \end{aligned}$$

Assumption (A-vect). Given a bounded, open, Lipschitz set $\Omega \subset {{\mathbb {R}}^{n}}$, $n\ge 2$, we investigate solutions ${{\varvec{\textsf {u}}}}:\Omega \rightarrow {{\mathbb {R}}^{m}}$ to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -{{\varvec{{\mathsf{\textrm{div}}}}}}{{\mathcal {A}}}(x,{D{\varvec{\textsf {u}}}})={{\varvec{{\mathsf{\mu }}}}}\quad \text {in }\ \Omega ,\\ {{\varvec{\textsf {u}}}}=0\quad \text {on }\ \partial \Omega \end{array}\right. } \end{aligned}$$

(4)

with a datum ${{\varvec{{\mathsf{\mu }}}}}$ being a vector-valued bounded Radon measure and a function ${{\mathcal {A}}}:\Omega \times {{\mathbb {R}}^{n\times m}}\rightarrow {{\mathbb {R}}^{n\times m}}$ is a weighted operator of Orlicz growth expressed by the means of $g(t):=G'(t)$, where an N-function $G \in C^2((0,\infty ))\cap C({[0,\infty )})$ satisfies $i_G\ge 2$ with $i_G$ given by (2). Let $g\in \Delta _2\cap \nabla _2$. Namely, ${{\mathcal {A}}}$ is assumed to admit a form

$$\begin{aligned} {{\mathcal {A}}}(x,\xi )=a(x)\frac{g(|\xi |)}{|\xi |}\,\xi , \end{aligned}$$

(5)

where $a:\Omega \rightarrow [c_a,C_a],$ $0<c_a<C_a$ is a continuous function with a modulus of continuity $\omega _a$. We define a potential

$$\begin{aligned} {\mathcal {W}}^{\mu }_G(x_0,R)=\int _0^R g^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B_r(x_0))}{r^{n-1}}\right) \,dr. \end{aligned}$$

(6)

For the case of referring to the dependence of some quantities on the parameters of the problem, we collect them as

$$\begin{aligned} {{\texttt {\textit{data}}}}={{\texttt {\textit{data}}}}(i_G,s_G,c_a,C_a,\omega _a,n,m). \end{aligned}$$

Having (5), one can infer the strong monotonicity of the vector field ${{\mathcal {A}}}$ of a form given by Lemma 3.2.

Let us define the notion of very weak solutions we employ. Let us define the notion of very weak solutions we employ, cf. [23].

Solutions Obtained as a Limit of Approximation. A map ${\varvec{\textsf {u}}}\in W^{1,1}_0(\Omega ,{{\mathbb {R}}^{m}})$ such that $\int _\Omega g(|{D{\varvec{\textsf {u}}}}|)\,dx<\infty$ is called a SOLA to (4) under the regime of Assumption (A-vect), if there exists a sequence $({\varvec{\textsf {u}}}_{h})\subset W^{1,G }(\Omega ,{{\mathbb {R}}^{m}})$ of local energy solutions to the systems

$$\begin{aligned} -{{\varvec{{\mathsf{\textrm{div}}}}}}{{\mathcal {A}}}(x,D{\varvec{\textsf {u}}}_{h})={{{\varvec{{\mathsf{\mu }}}}}_h} \end{aligned}$$

such that ${\varvec{\textsf {u}}}_{h}\rightarrow {\varvec{\textsf {u}}}$ locally in $W^{1,1}(\Omega ,{{\mathbb {R}}^{m}})$ and $({{\varvec{{\mathsf{\mu }}}}}_h)\subset L^\infty (\Omega ,{{\mathbb {R}}^{m}})$ is a sequence of maps that converges to ${{\varvec{{\mathsf{\mu }}}}}$ weakly in the sense of measures and satisfies

$$\begin{aligned} \limsup _h |{{\varvec{{\mathsf{\mu }}}}}_h |(B) \le |{{\varvec{{\mathsf{\mu }}}}}|(B) \end{aligned}$$

(7)

for every ball $B\subset \Omega$.

Observe that the above approximation property immediately implies that a SOLA ${\varvec{\textsf {u}}}$ is a distributional solution to (4), that is,

$$\begin{aligned} \int _\Omega {{\mathcal {A}}}(x,{D{\varvec{\textsf {u}}}}):D{{{\varvec{{\mathsf{\varphi }}}}}}\,dx=\int {{\varvec{{\mathsf{\varphi }}}}}\,d{{\varvec{{\mathsf{\mu }}}}}\qquad \text {for every }{{\varvec{{\mathsf{\varphi }}}}}\in C^\infty (\Omega ,{{\mathbb {R}}^{m}}). \end{aligned}$$

2.2 Main results

Our main accomplishment reads as follows.

Theorem 2.1

(Pointwise Wolff potential estimates) Suppose ${\varvec{\textsf {u}}}:\Omega \rightarrow {{\mathbb {R}}^{m}}$ is a SOLA to (4) with ${{\mathcal {A}}}$ satisfying Assumption (A-vect) and ${{\varvec{{\mathsf{\mu }}}}}\in {{\mathcal {M}}(\Omega ,{{\mathbb {R}}^{m}})}$. Let $B_r(x_0)\Subset \Omega$ with $r<R_0$ for some $R_0=R_0({{\texttt {\textit{data}}}})$. If ${{\mathcal {W}}}^{{\varvec{{\mathsf{\mu }}}}}_G(x_0,r)$ is finite, then $x_0$ is a Lebesgue’s point of ${\varvec{\textsf {u}}}$ and

(8)

holds for $C_{{\mathcal {W}}}>0$ depending only on ${{\texttt {\textit{data}}}}$. In particular, we have the following pointwise estimate

(9)

2.3 Local behaviour of very weak solutions

Potential estimates are known to be efficient tools to bring precise information on the local behaviour of solutions. We refer to [62] for clearly presented overview of consequences of estimates like (9) in studies on p-superharmonic functions together with a bunch of related references and to [22] for similar results for ${\mathcal {A}}$-harmonic functions where the operator ${\mathcal {A}}$ is exhibiting Orlicz type of growth. Notice however we referred to the scalar case. In the vectorial one, the only investigations on the potential estimates to solutions to measure data problem we are aware of are available in [27, 65] for p-Laplace systems. Let us present the regularity consequences of Wolff potential estimates to p-Laplace system with continuous coefficients and the natural scope of operators with similar structure and Orlicz growth.

We start with finding a density condition around a point $x_0$ implying that a solution has vanishing mean oscillations at $x_0$. The proposition below does not follow from Theorem 2.1, for the proof see Sect. 5.

Proposition 2.2

(VMO criterion) Suppose ${{\mathcal {A}}}$ satisfies Assumption (A-vect) and ${{\varvec{{\mathsf{\mu }}}}}\in {{\mathcal {M}}(\Omega ,{{\mathbb {R}}^{m}})}$. Let ${\varvec{\textsf {u}}}$ be a SOLA to (4) and let $B_r(x_0)\Subset \Omega .$ If

$$\begin{aligned} \lim _{\varrho \rightarrow 0}\varrho \, g^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B_\varrho (x_0))}{\varrho ^{n-1}}\right) =0, \end{aligned}$$

(10)

then ${\varvec{\textsf {u}}}$ has vanishing mean oscillations at $x_0$, i.e.

(11)

An application of Theorem 2.1 is the following continuity criterion proven also in Sect. 5.

Theorem 2.3

(Continuity criterion) Suppose ${\varvec{\textsf {u}}}$ is a SOLA to (4) under the regime of Assumption (A-vect) and $B_r(x_0)\Subset \Omega .$ If

$$\begin{aligned} \lim _{\varrho \rightarrow 0}\sup _{x\in B_r(x_0)}{{\mathcal {W}}}^{{\varvec{{\mathsf{\mu }}}}}_G(x,\varrho )=0, \end{aligned}$$

(12)

then ${\varvec{\textsf {u}}}$ is continuous in $B_r(x_0).$

If ${{\varvec{{\mathsf{\mu }}}}}=0$, then ${{\mathcal {W}}}^{{\varvec{{\mathsf{\mu }}}}}_G(x,\varrho )=0$, thus trivially we have the following consequence.

Corollary 2.4

Under Assumption (A-vect) if ${\varvec{\textsf {u}}}$ is an ${{\mathcal {A}}}$-harmonic map in $\Omega ,$ then ${\varvec{\textsf {u}}}$ is continuous in every $\Omega '\Subset \Omega .$

Condition (12) holds true provided the datum belongs to a Lorentz-type space. In order to define it we recall some definitions. We denote by $f^\star$ the decreasing rearrangement of a measurable function $f:\Omega \rightarrow {{\mathbb {R}}}$ by

$$\begin{aligned} f^\star (t) = \sup \{ s \ge 0 :|\{x\in {{\mathbb {R}}}^n:f(x)>s\}| > t \}, \end{aligned}$$

the maximal rearrangement by

$$\begin{aligned} f^{\star \star }(t) =\frac{1}{t} \int _0^t f^\star (s) \, ds\quad \text {and}\quad f^{\star \star }(0)= f^{\star }(0). \end{aligned}$$

Following [80] by Lorentz space $L(\alpha , \beta )(\Omega )$ for $\alpha ,\beta >0$ we mean the class of measurable functions such that

$$\begin{aligned} \int _0^\infty \left( t^{1/ \alpha } f^{\star \star }(t)\right) ^\beta \,\frac{dt}{t} <\infty . \end{aligned}$$

The following fact is proven in Appendix.

Lemma 2.5

Suppose ${{\varvec{{\mathsf{\mu }}}}}={\varvec{\textsf {F}}}:{{\mathbb {R}}^{n}}\rightarrow {{\mathbb {R}}^{m}}$ is a locally integrable map vanishing outside $\Omega$, then there exists a constant $c=c(n,i_G,s_G)>0$, such that

$$\begin{aligned} {{\mathcal {W}}}_G^{{\varvec{\textsf {F}}}}(x,R)\le c\int _0^{|B_R|} t^\frac{1}{n} g^{-1}\left( t^\frac{1}{n} |{\varvec{\textsf {F}}}|^{\star \star }(t)\right) \,\frac{dt}{t}=: {\mathcal {I}}_R. \end{aligned}$$

Note that if ${\mathcal {I}}_R<\infty$, the datum ${\varvec{\textsf {F}}}$ is in the dual space to $W^{1,G}(B_R)$ and, consequently, we deal with a weak solution, see [3]. Let us present a corollary of Theorem 2.3 holding due to Lemma 2.5.

Corollary 2.6

If ${\varvec{\textsf {u}}}$ is a weak solution to $-{{\varvec{{\mathsf{\textrm{div}}}}}}{{\mathcal {A}}}(x,{D{\varvec{\textsf {u}}}})={\varvec{\textsf {F}}}$ with ${{\mathcal {A}}}$ satisfying Assumption (A-vect) and ${\varvec{\textsf {F}}}:\Omega \rightarrow {{\mathbb {R}}^{m}}$ such that

$$\begin{aligned} {\mathcal {I}}:=\int _0^{|\Omega |} t^\frac{1}{n} g^{-1}\left( t^\frac{1}{n} |{\varvec{\textsf {F}}}|^{\star \star }(t)\right) \,\frac{dt}{t} <\infty \end{aligned}$$

(13)

for $\Omega _0\Subset \Omega$, then ${\varvec{\textsf {u}}}\in C(\Omega _0,{{\mathbb {R}}^{m}})$ and $\Vert {\varvec{\textsf {u}}}\Vert _{L^\infty (\Omega _0,{{\mathbb {R}}^{m}})}\le c({{\texttt {\textit{data}}}}) {\mathcal {I}}.$ This bound is optimal and attained by a radial solution on a ball, see [3]. Moreover, as a special case we get that ${\varvec{\textsf {u}}}$ is continuous under the regularity restrictions on $|{\varvec{\textsf {F}}}|=f$ of [3, Example 1 (A) and Example 2 (A)], still within our regime requiring $g\in \Delta _2\cap \nabla _2$ and $i_G\ge 2$.

The above corollary results in the following extension of [65, Theorem 10.6] to the weighted case.

Remark 2.7

If ${\varvec{\textsf {u}}}$ is a weak solution to $-{ {{\varvec{{\mathsf{\textrm{div}}}}}}}\left( a(x)|{D{\varvec{\textsf {u}}}}|^{p-2} {D{\varvec{\textsf {u}}}}\right) ={\varvec{\textsf {F}}}$ for $p\ge 2$ and $0<a\in C(\Omega )$ is separated from zero and $|{\varvec{\textsf {F}}}|$ belongs locally to the Lorentz space $L(\tfrac{n}{p},\tfrac{1}{p-1})(\Omega )$ then ${\varvec{\textsf {u}}}$ is continuous in $\Omega$.

As another application of Corollary 2.6 let us present its consequences for the Zygmund case.

Remark 2.8

Suppose that $2\le p<n,$ $\alpha \ge 0$, $0<a\in C(\Omega )$ separated from zero is bounded, and ${\varvec{\textsf {u}}}$ is a weak solution to

$$\begin{aligned} -{ {{\varvec{{\mathsf{\textrm{div}}}}}}}\left( a(x)|{D{\varvec{\textsf {u}}}}|^{p-2}\log ^\alpha (\textrm{e}+|{D{\varvec{\textsf {u}}}}|) \,{D{\varvec{\textsf {u}}}}\right) ={\varvec{\textsf {F}}}. \end{aligned}$$

(14)

Observe that in this case $g^{-1}(\lambda )\approx \lambda ^\frac{1}{p-1}\log ^{-\frac{\alpha }{p-1}}(\textrm{e}+\lambda )$. If ${\varvec{\textsf {F}}}$ satisfies (13), then ${\varvec{\textsf {u}}}$ is continuous in $\Omega$.

Wolff potential estimates can be used to find a relevant condition on a measure ${{\varvec{{\mathsf{\mu }}}}}$ to infer Hölder continuity of solutions. One of the natural ones is expressed in the Orlicz modification of the Morrey-type scale.

Theorem 2.9

(Hölder continuity criterion) Suppose ${\varvec{\textsf {u}}}$ is a SOLA to (4) under the regime of Assumption (A-vect). Assume further that for ${{\varvec{{\mathsf{\mu }}}}}$ there exist positive constants $c=c({{\texttt {\textit{data}}}})>0$ and $\theta \in (0,1)$ such that

$$\begin{aligned} |{{\varvec{{\mathsf{\mu }}}}}|(B_r(x))\le c r^{n-1}g(r^{\theta -1})\approx r^{n-\theta }G(r^{\theta -1}) \end{aligned}$$

(15)

for each $B_r(x)\Subset \Omega$ with sufficiently small radius. Then ${\varvec{\textsf {u}}}$ is locally Hölder continuous in $\Omega$.

For p-growth problems condition (15) reads as $|{{\varvec{{\mathsf{\mu }}}}}|(B_r(x))\le c r^{n-p+\theta (p-1)}$ well known since [14, 57, 59, 78]. Moreover, in the scalar case (15) is proven in [22] to be equivalent to Hölder continuity of solutions, while in [20] to characterize removable sets for Hölder continuous solutions. In the vectorial case we cannot get equivalence, because by Theorem 2.1 we are equipped with one-sided estimate only. Specializing Theorem 2.9, we have the following results.

Remark 2.10

Suppose $p\ge 2$, positive $a\in C(\Omega )$ is separated from zero, and ${\varvec{\textsf {u}}}$ is a SOLA to

$$\begin{aligned} -{ {{\varvec{{\mathsf{\textrm{div}}}}}}}\left( a(x)|{D{\varvec{\textsf {u}}}}|^{p-2} {D{\varvec{\textsf {u}}}}\right) ={{\varvec{{\mathsf{\mu }}}}}\end{aligned}$$

with $|{{\varvec{{\mathsf{\mu }}}}}|(B_r(x))\le c r^{n-p+\theta (p-1)}$ for some $c>0,$ $\theta \in (0,1)$ and all sufficiently small $r>0.$ Then ${\varvec{\textsf {u}}}$ is locally Hölder continuous in $\Omega$.

Remark 2.11

Suppose $p\ge 2,$ $\alpha \in {{\mathbb {R}}}$, positive $a\in C(\Omega )$ is separated from zero, and ${\varvec{\textsf {u}}}$ is a SOLA to

$$\begin{aligned} - {{\varvec{{\mathsf{\textrm{div}}}}}}\left( a(x)|{D{\varvec{\textsf {u}}}}|^{p-2}\log ^\alpha (\textrm{e}+|{D{\varvec{\textsf {u}}}}|) {D{\varvec{\textsf {u}}}}\right) ={{\varvec{{\mathsf{\mu }}}}}\end{aligned}$$

(16)

with $|{{\varvec{{\mathsf{\mu }}}}}|(B_r(x))\le c r^{n-p+\theta (p-1)}\log ^\alpha (\textrm{e}+r^{\theta -1})$ for some $c>0,$ $\theta \in (0,1)$ and all sufficiently small $r>0.$ Then ${\varvec{\textsf {u}}}$ is locally Hölder continuous in $\Omega$.

The sufficient condition for (15) and, in turn, for the Hölder continuity of the solution is to assume that $|{{\varvec{{\mathsf{\mu }}}}}|=|{\varvec{\textsf {F}}}|$ belongs to a relevant Marcinkiewicz-type space. Following [77] for a continuous increasing function $\psi :(0,|\Omega |)\rightarrow (0,\infty )$ we say that $f\in L(\psi ,\infty )(\Omega )$ if the maximal rearrangement $f^{\star \star }$ of f satisfies

$$\begin{aligned} \sup _{s\in (0,|\Omega |)}\frac{f^{\star \star }(s)}{\psi ^{-1}(1/s)}<\infty . \end{aligned}$$

There holds the following consequence of Theorem 2.9.

Corollary 2.12

Suppose ${\varvec{\textsf {u}}}$ is a SOLA to $-{{\varvec{{\mathsf{\textrm{div}}}}}}{{\mathcal {A}}}(x,{D{\varvec{\textsf {u}}}})={\varvec{\textsf {F}}}$ under the regime of Assumption (A-vect) and $|{\varvec{\textsf {F}}}|$ belongs locally to the Marcinkiewicz-type space $L(\psi ,\infty )(\Omega )$ with $\psi ^{-1}(1/\lambda )=\lambda ^{-\frac{1}{n}}g\big (\lambda ^{\frac{\theta -1}{n}}\big )$ for some $\theta \in (0,1)$, then ${\varvec{\textsf {u}}}$ is locally Hölder continuous.

For justification that indeed under the assumptions of Corollary 2.12 the condition (15) is satisfied see the calculations provided for the scalar case [22, Sect. 2]. The above fact has the best possible consequence in the p-Laplace case.

Remark 2.13

If $p\ge 2$, positive $a\in C(\Omega )$ is separated from zero, ${\varvec{\textsf {u}}}$ is a SOLA to $-{ {{\varvec{{\mathsf{\textrm{div}}}}}}}\left( a(x)|{D{\varvec{\textsf {u}}}}|^{p-2} {D{\varvec{\textsf {u}}}}\right) ={\varvec{\textsf {F}}}$ and $|{\varvec{\textsf {F}}}|$ belongs locally to the Marcinkiewicz space $L(\frac{n}{p+\theta (p-1)},\infty )(\Omega )$ for some $\theta \in (0,1)$, i.e. $\sup _{\lambda>0}\left( \lambda ^\frac{n}{p+\theta (p-1)}\big |\{x\in \Omega _0:\,|{\varvec{\textsf {F}}}(x)|>\lambda \}|\right) <\infty$ for any $\Omega _0\Subset \Omega$, then ${\varvec{\textsf {u}}}$ is locally Hölder continuous in $\Omega$.

Remark 2.14

When $G(t)\approx t^p\log ^\alpha (\textrm{e}+t),$ $p\ge 2,$ $\alpha \in {{\mathbb {R}}}$, ${\varvec{\textsf {u}}}$ is a SOLA to (14) with ${\varvec{\textsf {F}}}$ such that

$$\begin{aligned} \sup _{\lambda>0}\left( \lambda ^\frac{n}{p+\theta (p-1)}\log ^{-\frac{\alpha (1-\theta )}{p+\theta (p-1)}}(\textrm{e}+\lambda ^\frac{n}{1-\theta })\big |\{x\in \Omega _0:\,|{\varvec{\textsf {F}}}(x)|>\lambda \}|\right) <\infty , \end{aligned}$$

for any $\Omega _0\Subset \Omega$, then ${\varvec{\textsf {u}}}$ is locally Hölder continuous in $\Omega$.

3 Preliminaries

3.1 Notation

We shall adopt the customary convention of denoting by c a constant that may vary from line to line. To skip rewriting a constant, we use $\lesssim$. By $a\approx b$, we mean $a\lesssim b$ and $b\lesssim a$. To stress the dependence of the intrinsic constants on the parameters of the problem, we write $\lesssim _{{\texttt {\textit{data}}}}$ or $\approx _{{\texttt {\textit{data}}}}$. By $B_R$ we denote a ball skipping prescribing its center, when it is not important. By $cB_R=B_{cR}$ we mean a ball with the same center as $B_R$, but with rescaled radius cR. We make use of the truncation operator, $T_k:{{\mathbb {R}}^{m}}\rightarrow {{\mathbb {R}}^{m}}$, defined as follows

$$\begin{aligned} T_k({{\varvec{{\mathsf{\xi }}}}}):=\min \left\{ 1,\frac{k}{|{{\varvec{{\mathsf{\xi }}}}}|}\right\} {{\varvec{{\mathsf{\xi }}}}}. \end{aligned}$$

(17)

Then, of course, $D T_k:{{\mathbb {R}}^{m}}\rightarrow {{\mathbb {R}}}^{m\times m}$ is given by

$$\begin{aligned} DT_k({{\varvec{{\mathsf{\xi }}}}})= {\left\{ \begin{array}{ll} {\textsf{Id}}&{}\text {if }\ |{{\varvec{{\mathsf{\xi }}}}}|\le k,\\ \frac{k}{|{{\varvec{{\mathsf{\xi }}}}}|}\left( {\textsf{Id}}-\frac{{{\varvec{{\mathsf{\xi }}}}}\otimes {{\varvec{{\mathsf{\xi }}}}}}{|{{\varvec{{\mathsf{\xi }}}}}|^2}\right) &{}\text {if }\ |{{\varvec{{\mathsf{\xi }}}}}|> k. \end{array}\right. } \end{aligned}$$

(18)

For a measurable set $U\subset {{\mathbb {R}}^{n}}$ with finite and positive n-dimensional Lebesgue measure $|U|>0$ and ${\varvec{\textsf {f}}}:U\rightarrow {\mathbb {R}}^{k}$, $k\ge 1$ being a measurable map, we define

By $C^{0,\gamma }(U)$, $\gamma \in (0,1]$, we mean the family of Hölder continuous functions, i.e. measurable functions $f:U\rightarrow {\mathbb {R}}^k$ for which

$$\begin{aligned}{}[{\varvec{\textsf {f}}}]_{0,\gamma }:=\sup _{\begin{array}{c} x,y\in U,\\ x\not =y \end{array}}\frac{|{\varvec{\textsf {f}}}(x)-{\varvec{\textsf {f}}}(y)|}{|{x-y}|^{\gamma }}<\infty . \end{aligned}$$

We describe the ellipticity of a vector field ${{\mathcal {A}}}$ using a function ${{\mathcal {V}}}:{{\mathbb {R}}^{n\times m}}\rightarrow {{\mathbb {R}}^{n\times m}}$ given by

$$\begin{aligned} {{\mathcal {V}}}({\xi })= \left( \frac{g(|{\xi }|)}{|{\xi }|}\right) ^{1/2}\xi . \end{aligned}$$

(19)

3.2 Basic definitions

References for this section are [61, 79].

We say that a function $G: [0, \infty ) \rightarrow [0, \infty ]$ is a Young function if it is convex, vanishes at 0, and is neither identically equal to 0, nor to infinity. A Young function G which is finite-valued, vanishes only at 0 and satisfies the additional growth conditions

$$\begin{aligned} \lim _{t \rightarrow 0}\frac{G (t)}{t}=0 \qquad \hbox {and} \qquad \lim _{t \rightarrow \infty }\frac{G (t)}{t}=\infty \end{aligned}$$

is called an N-function. The complementary function $\widetilde{G}$ (called also the Young conjugate, or the Legendre transform) to a nondecreasing function $G:{[0,\infty )}\rightarrow {[0,\infty )}$ is given by the following formula

$$\begin{aligned} \widetilde{G}(s):=\sup _{t>0}(s\cdot t-G(t)). \end{aligned}$$

If G is a Young function, so is $\widetilde{G}$. If G is an N-function, so is $\widetilde{G}$.

Having Young functions $G,\widetilde{G}$, we are equipped with Young’s inequality reading as follows

$$\begin{aligned} ts\le G(t)+\widetilde{G}(s)\quad \text {for all }\ s,t\ge 0. \end{aligned}$$

(20)

We say that a function $G:{[0,\infty )}\rightarrow {[0,\infty )}$ satisfies $\Delta _2$-condition if there exists $c_{\Delta _2}>0$ such that $G(2t)\le c_{\Delta _2}G(t)$ for $t>0.$ We say that G satisfy $\nabla _2$-condition if $\widetilde{G}\in \Delta _2.$ Note that it is possible that G satisfies only one of the conditions $\Delta _2/\nabla _2$. For instance, for $G(t) =( (1+|t|)\log (1+|t|)-|t|)\in \Delta _2$, its complementary function is ${\widetilde{G}}(s)= (\exp (|s|)-|s|-1 )\not \in \Delta _2$. See [79, Sect. 2.3, Theorem 3] for equivalence of various definitions of these conditions and [21, 31] for illustrating the subtleties. In particular, $G\in \Delta _2\cap \nabla _2$ if and only if $1<i_G\le s_G<\infty$, see (2). This assumption implies a comparison with power-type functions i.e. $\frac{G(t)}{t^{i_G}}$ is non-decreasing and $\frac{G(t)}{t^{s_G}}$ is non-increasing, but it is stronger than being sandwiched between power functions.

Lemma 3.1

If an N-function $G\in \Delta _2\cap \nabla _2$, then $g(t)t\approx G(t)$

and $\widetilde{G}(g(t))\approx G(t)$ with the constants depending only on the growth indexes of G, that is $i_G$ and $s_G$. Moreover, $g^{-1}(2t)\le c g^{-1}(t)$ with $c=c(i_G,s_G).$

Due to Lemma 3.1 and [33, Lemmas 3 and 21], we have the following relations.

Lemma 3.2

If G is an N-function of class $C^2((0,\infty ))\cap C([0,\infty ))$, $G{,g}\in \Delta _2\cap \nabla _2$, ${{\mathcal {A}}}$ is given by (5), then for every $\xi ,\eta \in {{\mathbb {R}}^{n\times m}}$ it holds

$$\begin{aligned} \left( {{\mathcal {A}}}(x,\xi )-{{\mathcal {A}}}(x,\eta )\right) :{(\xi -\eta )}\gtrsim _{{{\texttt {\textit{data}}}}}\, \frac{g(|\xi |+|\eta |)}{|\xi |+|\eta |}|\xi -\eta |^2\approx _{{{\texttt {\textit{data}}}}} \left| {{\mathcal {V}}}(\xi )-{{\mathcal {V}}}(\xi ) \right| ^2,\qquad \end{aligned}$$

(21)

and

$$\begin{aligned} g(|\xi |+|\eta |)|\xi -\eta |\approx _{{{\texttt {\textit{data}}}}} G^\frac{1}{2}(|\xi |+|\eta |) |{{\mathcal {V}}}(\xi )-{{\mathcal {V}}}(\eta )|. \end{aligned}$$

(22)

3.3 Orlicz spaces

Basic reference for this section is [2], where the theory of Orlicz spaces is presented for scalar functions. The proofs for functions with values in ${{\mathbb {R}}^{m}}$ can be obtained by obvious modifications.

We study the solutions to PDEs in the Orlicz-Sobolev spaces equipped with a modular function $G\in C^1 {((0,\infty ))}$ - a strictly increasing and convex function such that $G(0)=0$ and satisfying (2). Let us define a modular

$$\begin{aligned} \varrho _{G,U}({{\varvec{{\mathsf{\xi }}}}})=\int _U G(|{{\varvec{{\mathsf{\xi }}}}}|)\,dx. \end{aligned}$$

(23)

For any bounded $\Omega \subset {{\mathbb {R}}^{n}}$, by Orlicz space ${L}^G(\Omega ,{{\mathbb {R}}^{m}})$ we understand the space of measurable functions endowed with the Luxemburg norm

$$\begin{aligned} ||{\varvec{\textsf {f}}}||_{L^G(\Omega , {{\mathbb {R}}^{m}})}=\inf \left\{ \lambda >0:\ \ \varrho _{G,\Omega }\left( \tfrac{1}{\lambda } |{\varvec{\textsf {f}}}|\right) \le 1\right\} . \end{aligned}$$

We define the Orlicz-Sobolev space $W^{1,G}(\Omega )$ as follows

$$\begin{aligned} W^{1,G}(\Omega ,{{\mathbb {R}}^{m}})=\big \{{\varvec{\textsf {f}}}\in {W^{1,1}(\Omega ,{{\mathbb {R}}^{m}})}:\ \ |{\varvec{\textsf {f}}}|,|D{{\varvec{\textsf {f}}}}|\in L^G(\Omega ,{{\mathbb {R}}^{m}})\big \}, \end{aligned}$$

where the gradient is understood in the distributional sense, endowed with the norm

$$\begin{aligned} \Vert {\varvec{\textsf {f}}}\Vert _{W^{1,G}(\Omega ,{{\mathbb {R}}^{m}})}=\inf \bigg \{\lambda >0 :\ \ \varrho _{G,\Omega }\left( \tfrac{1}{\lambda } |{\varvec{\textsf {f}}}|\right) + \varrho _{G,\Omega }\left( \tfrac{1}{\lambda } |D{{\varvec{\textsf {f}}}}|\right) \le 1\bigg \} \end{aligned}$$

and by $W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ we denote the closure of $C_c^\infty (\Omega ,{{\mathbb {R}}^{m}})$ under the above norm. Since condition (2) imposed on G implies $G,\widetilde{G}\in \Delta _2$, the Orlicz-Sobolev space $W^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ we deal with is separable and reflexive. Moreover, one can apply arguments of [49] to infer density of smooth functions in $W^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$.

The counterpart of the Hölder inequality in this setting reads

$$\begin{aligned} \Vert {\varvec{\textsf {f}}}{\varvec{\textsf {g}}}\Vert _{L^1(\Omega ,{{\mathbb {R}}^{m}})}\le 2\Vert {\varvec{\textsf {f}}}\Vert _{L^G(\Omega ,{{\mathbb {R}}^{m}})}\Vert {\varvec{\textsf {g}}}\Vert _{L^{\widetilde{G}}(\Omega ,{{\mathbb {R}}^{m}})} \end{aligned}$$

(24)

for all ${\varvec{\textsf {f}}}\in L^G(\Omega ,{{\mathbb {R}}^{m}})$ and ${\varvec{\textsf {g}}}\in L^{\widetilde{G}}(\Omega ,{{\mathbb {R}}^{m}})$.

3.4 The operator

We notice that in such regime the operator ${\textsf{A}}_{G}$ acting as

$$\begin{aligned} \langle {\textsf{A}}_{G}{\varvec{\textsf {u}}},{{\varvec{{\mathsf{\phi }}}}}\rangle :=\int _{\Omega }{{\mathcal {A}}}(x,{D{\varvec{\textsf {u}}}}): {D{{\varvec{{\mathsf{\phi }}}}}}\, dx\quad \text {for}\quad {{\varvec{{\mathsf{\phi }}}}}\in C^{\infty }_{0}(\Omega ,{{\mathbb {R}}^{m}}) \end{aligned}$$

is well defined on a reflexive and separable Banach space $W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ and ${\textsf{A}}_{G}(W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}}))\subset (W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}}))'$. Indeed, when ${\varvec{\textsf {u}}}\in W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ and ${{\varvec{{\mathsf{\phi }}}}}\in C_c^\infty (\Omega ,{{\mathbb {R}}^{m}})$, structure condition (5), Hölder’s inequality (24), and Lemma 3.1 justify that

$$\begin{aligned} |{\langle {\textsf{A}}_{G}{\varvec{\textsf {u}}},{{\varvec{{\mathsf{\phi }}}}}\rangle }|\le&\,c\,\int _{\Omega }g(|{{D{\varvec{\textsf {u}}}}}|){|{D{{\varvec{{\mathsf{\phi }}}}}}|} \, dx \le c\left\| g(|{{D{\varvec{\textsf {u}}}}}|) \right\| _{L^{\widetilde{G}(\cdot )} }\Vert {|{D{{\varvec{{\mathsf{\phi }}}}}}|}\Vert _{L^{G}} \\ \le&\, c\Vert {|{D{\varvec{\textsf {u}}}}|}\Vert _{L^{ G}}\Vert {|{D{{\varvec{{\mathsf{\phi }}}}}}|}\Vert _{L^{G}}\le c\Vert {{{\varvec{{\mathsf{\phi }}}}}}\Vert _{W^{1,G}}. \end{aligned}$$

3.5 Definitions of solutions and comments on existence results

We stress that the problems are considered under the regime of Assumption (A-vect).

A function ${\varvec{\textsf {v}}}\in W_{loc}^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ is called an $\underline{{{\mathcal {A}}}}$ -harmonic map in $\Omega \subset {{\mathbb {R}}^{n}}$ provided

$$\begin{aligned} \int _\Omega {{\mathcal {A}}}(x,{D{\varvec{\textsf {v}}}}):D{{{\varvec{{\mathsf{\varphi }}}}}}\,dx=0\qquad \text {for all }\ {{\varvec{{\mathsf{\varphi }}}}}\in C_c^\infty (\Omega ,{{\mathbb {R}}^{m}}). \end{aligned}$$

(25)

As a consequence of our main result, in Corollary 2.4, we prove that ${{\mathcal {A}}}$-harmonic maps are continuous. In fact, by Campanato’s characterization [48, Theorem 2.9] one can infer from Proposition 3.13 Hölder continuity $C^{0,\gamma }(\Omega ,{{\mathbb {R}}^{m}})$ of ${{\mathcal {A}}}$-harmonic maps with any exponent $\gamma \in (0,1).$

A function ${\varvec{\textsf {u}}}\in W^{1,G}_{loc}(\Omega ,{{\mathbb {R}}^{m}})$ is called a weak solution to (4), if

$$\begin{aligned} \int _\Omega {{\mathcal {A}}}(x,{D{\varvec{\textsf {u}}}}): D{{{\varvec{{\mathsf{\varphi }}}}}}\,dx=\int _\Omega {{\varvec{{\mathsf{\varphi }}}}}\,d{{\varvec{{\mathsf{\mu }}}}}(x)\quad \text {for every }\ {{\varvec{{\mathsf{\varphi }}}}}\in C^\infty _c(\Omega ,{{\mathbb {R}}^{m}}). \end{aligned}$$

(26)

Recall that $W^{1,G}_{0}(\Omega ,{{\mathbb {R}}^{m}})$ is separable and by its definition ${C_c^\infty }(\Omega ,{{\mathbb {R}}^{m}})$ is dense there.

Remark 3.3

(Existence and uniqueness of weak solutions) For ${{\varvec{{\mathsf{\mu }}}}}\in (W^{1,G}_{0}(\Omega ,{{\mathbb {R}}^{m}}))',$ due to the strict monotonicity of the operator, there exists a unique weak solution to (4), see [56, Sect. 3.1].

Recall that the notion of SOLA is defined in Sect. 2.2. The problem (4) admits a solution of this type for arbitrary bounded measure.

Proposition 3.4

If a vector field ${{\mathcal {A}}}$ satisfies Assumption (A-vect) and ${{\varvec{{\mathsf{\mu }}}}}\in {{\mathcal {M}}(\Omega ,{{\mathbb {R}}^{m}})}$, then there exists a SOLA ${\varvec{\textsf {u}}}\in W^{1,1}_0(\Omega ,{{\mathbb {R}}^{m}})$ to (4).

The idea to prove it is to consider

$$\begin{aligned} {\varvec{\textsf {f}}}_k(x):= \int _{{\mathbb {R}}^{m}}\varrho _k(x-y)\, d{{\varvec{{\mathsf{\mu }}}}}(y), \end{aligned}$$

where $\varrho _k$ stands for a standard mollifier i.e. for a nonnegative, smooth, and even function such that $\int _{{\mathbb {R}}^{m}}\varrho (s)\mathrm{\,d}s=1$ we define $\varrho _k(s)=k^n\varrho (ks)$ for $k\in {{\mathbb {N}}}$. Of course

By Remark 3.3 one finds ${\varvec{\textsf {u}}}_k\in W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ such that for every ${{\varvec{{\mathsf{\varphi }}}}}\in W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ it holds that

$$\begin{aligned} \int _\Omega {{\mathcal {A}}}(x,{D{\varvec{\textsf {u}}}}_k): D{{{\varvec{{\mathsf{\varphi }}}}}}\,dx=\int _\Omega {{\varvec{{\mathsf{\varphi }}}}}\,{\varvec{\textsf {f}}}_k dx. \end{aligned}$$

The existence of SOLA by passing to the limit can be justified by modification of arguments of [39].

See [3, 8, 18, 21, 24, 26] for related existence and regularity results in the scalar case and [28, 39, 40, 46, 66] for vectorial existence results in various regimes.

3.6 Auxiliary results

Lemma 3.5

For $\varvec{\textsf {g}}:B\rightarrow {{\mathbb {R}}}^k$, $k\ge 1$ and any ${{\varvec{{\mathsf{\xi }}}}}\in {{\mathbb {R}}}^k$ it holds that

We have the following corollary of the Cavalieri Principle.

Lemma 3.6

If $\nu \in {\mathcal {M}}(\Omega )$ has a density $\omega$ (i.e. $d\nu =\omega (x)\,dx$ with $\omega \in L^1(\Omega )$) and $(1+|f|)^{-(\gamma +1)}\omega \in L^1({{\mathbb {R}}^{n}})$ for some $\gamma >0$, then

$$\begin{aligned} \int _0^\infty \frac{\nu (\{|f|<t\})}{(1+t)^{2+\gamma }}\,dt=\frac{1}{1+\gamma }\int _{{\mathbb {R}}^{n}}\frac{d\nu }{(1+|f|)^{\gamma +1}}. \end{aligned}$$

Lemma 3.7

([48], Lemma 6.1) Let $\phi :[R/2,3R/4]\rightarrow [0,\infty )$ be a function such that

$$\begin{aligned} \phi (r_1)\le \frac{1}{2}\phi (r_2)+{\textsf{A}}+\frac{{\textsf{B}}}{(r_2-r_1)^\beta }\qquad \text {for every}\qquad R/2\le r_1<r_2\le 3{R}/4 \end{aligned}$$

with $\textsf{A,B}\ge 0$ and $\beta >0$. Then there exists $c=c(\beta )$, such that

$$\begin{aligned} \phi (R/2)\le c \left( {\textsf{A}}+\frac{{\textsf{B}}}{R^\beta }\right) . \end{aligned}$$

Lemma 3.8

([50], Lemma 3.4) Let $\phi (t)$ be a nonnegative and nondecreasing function on [0, R]. Suppose that

$$\begin{aligned} \phi (\rho ) \le {\textsf{A}} \left[ \left( \frac{\rho }{r}\right) ^{\alpha } + \epsilon \right] \phi (r) + {\textsf{B}} r^{\beta } \end{aligned}$$

for any $0 < \rho \le r \le R$, with ${\textsf{A}},{\textsf{B}},\alpha ,\beta$ nonnegative constants and $\beta < \alpha$. Then for any $\gamma \in (\beta , \alpha )$, there exists a constant $\epsilon _{0} = \epsilon _{0}({\textsf{A}},\alpha , \beta , \gamma )$ such that if $\epsilon < \epsilon _0$, then for all $0 < \rho \le r \le R$ we have

$$\begin{aligned} \phi (\rho ) \le c \left\{ \left( \frac{\rho }{r} \right) ^{\gamma } \phi (r) + {\textsf{B}} \rho ^{\beta } \right\} \end{aligned}$$

where c is a positive constant depending on ${\textsf{A}}, \alpha , \beta , \gamma$.

The next lemma is a self-improving property for the reverse Hölder inequalities.

Lemma 3.9

([53], Lemma 3.38) Let $0<r<q<p<\infty$, $0< \rho < R \le 1$ and $w \in L^{p}(B_1)$. If the following reverse Hölder inequality holds

$$\begin{aligned} \bigg ( \int _{B_{\sigma '}} w^p \, dx \bigg )^{1/p} \le \frac{c_0}{(\sigma -\sigma ')^{\varkappa }} \bigg ( \int _{B_{\sigma }} w^{q} \, dx \bigg )^{1/q} + c_1 \end{aligned}$$

for some constants $c_0$ and $c_1$, whenever $\rho \le \sigma ' \le \sigma \le R$. Then there exists $c=c(c_0,\xi ,p,q,r)$ such that

$$\begin{aligned} \bigg ( \int _{B_{\rho }} w^p \, dx \bigg )^{1/p} \le \frac{c}{(R - \rho )^{\widetilde{\varkappa }}} \bigg ( \int _{B_{R}} w^{r} \, dx \bigg )^{1/r} + c_1, \end{aligned}$$

where

$$\begin{aligned} \widetilde{\varkappa }=\frac{\varkappa r(p-q)}{q(p-r)}. \end{aligned}$$

The following modular version of the Sobolev-Poincaré inequality follows almost directly as in [5], but we present the proof for vector-valued functions for the sake of completeness.

Proposition 3.10

Suppose $\Omega$ is a bounded Lipschitz domain in ${{\mathbb {R}}^{n}}$, $m,n\ge 1$, and $G:{[0,\infty )}\rightarrow {[0,\infty )}$ is an N-function such that $G\in \Delta _2\cap \nabla _2$. Then there exist a constant $C=C(n,m,|\Omega |,G)>0$, such that for every ${\varvec{\textsf {u}}}\in W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$

$$\begin{aligned} \int _\Omega G^{n'}(|{\varvec{\textsf {u}}}|)\,dx\le C\left( \int _\Omega G (|{D{\varvec{\textsf {u}}}}|)\,dx\right) ^{n'}. \end{aligned}$$

Proof

We provide the proof only in the case of continuously differentiable G. Otherwise every time one can find a sufficiently smooth function $G_\circ$ comparable to G, i.e. such that there exists $c>0$, such that $G_\circ (t)/c\le G(t)\le cG_\circ (t).$ Moreover, we start with the proof for fixed ${\varvec{\textsf {u}}}\in C_c^\infty (\Omega ,{{\mathbb {R}}^{m}})$ and then conclude by the density argument. The classical Sobolev inequality in $W^{1,1}$ gives

$$\begin{aligned} \left( \int _\Omega G^{n'}(|{\varvec{\textsf {u}}}|)\,dx\right) ^\frac{1}{n'}\le c\,\int _\Omega |{D(G(| {\varvec{\textsf {u}}}|))}|\,dx . \end{aligned}$$

(27)

Since $G\in \Delta _2,$ it satisfies $g(t)\le c\, G(t)/t$ and $G^*\left( \frac{G(t)}{t}\right) \le {G(t)}.$ Thus by the Young inequality we arrive at

$$\begin{aligned} \begin{aligned} |{D(G(| {\varvec{\textsf {u}}}|))}|&= g(|{\varvec{\textsf {u}}}|)\big |{D|{\varvec{\textsf {u}}}|}\big |\le c \frac{G(|{\varvec{\textsf {u}}}|)}{|{\varvec{\textsf {u}}}|}|{D{\varvec{\textsf {u}}}}|\\&\le \varepsilon G^*\left( \frac{G(|{\varvec{\textsf {u}}}|)}{|{\varvec{\textsf {u}}}|}\right) +c \, G(|{D{\varvec{\textsf {u}}}}|)\le \varepsilon G(|{\varvec{\textsf {u}}}|)+c\, G(|{D{\varvec{\textsf {u}}}}|). \end{aligned} \end{aligned}$$

(28)

Summing up, we have

$$\begin{aligned} \left( \int _\Omega G^{n'}(|{\varvec{\textsf {u}}}|)\,dx\right) ^\frac{1}{n'}\le C\varepsilon \int _\Omega G(| {\varvec{\textsf {u}}}|) \,dx+Cc_\varepsilon \int _\Omega G(|{D{\varvec{\textsf {u}}}}|)\,dx, \end{aligned}$$

where according to the Hölder inequality we obtain

$$\begin{aligned} \left( \int _\Omega G^{n'}(|{\varvec{\textsf {u}}}|)\,dx\right) ^\frac{1}{n'} \le \varepsilon C|\Omega |^\frac{1}{n}\left( \int _\Omega G^{n'} (| {\varvec{\textsf {u}}}|) \,dx\right) ^\frac{1}{n'}+Cc_\varepsilon \int _\Omega G(|{D{\varvec{\textsf {u}}}}|)\,dx. \end{aligned}$$

Now we can choose $\varepsilon$ small enough to absorb it on the right-hand side and obtain the claim for ${\varvec{\textsf {u}}}\in C_c^\infty (\Omega ,{{\mathbb {R}}^{m}})$. Since smooth function are dense in $W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ by standard approximation argument we get the claim for all ${\varvec{\textsf {u}}}\in W_0^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$. $\square$

3.7 Properties of ${{\mathcal {A}}}$-harmonic maps

Let us establish some fundamental properties of ${{\mathcal {A}}}$-harmonic maps.

Proposition 3.11

(Caccioppoli estimate) If ${\varvec{\textsf {v}}}\in W_0^{1,G} (\Omega ,{{\mathbb {R}}^{m}})$ is a nonnegative ${{\mathcal {A}}}$-harmonic map, ${{\varvec{{\mathsf{\lambda }}}}}\in {{\mathbb {R}}^{m}}$, and $7/8\le \sigma '<\sigma \le 1,$ then there exists $c=c({{\texttt {\textit{data}}}})>0$, such that

(29)

Proof

Let us pick a cutoff function $\eta \in C_c^\infty ({B_{\sigma r}})$ such that $\mathbb {1}_{B_{\sigma ' r}}\le \eta \le \mathbb {1}_{B_{\sigma r}}$ and $|D\eta |\le c_1/(\sigma '-\sigma )$. We use $\xi =\eta ^q ({\varvec{\textsf {v}}}-{{\varvec{{\mathsf{\lambda }}}}})$ as a test function to get

$$\begin{aligned} \int _\Omega {{\mathcal {A}}}(x,{D{\varvec{\textsf {v}}}}): {D{\varvec{\textsf {v}}}}\,\eta ^q \,dx =\int _\Omega {{\mathcal {A}}}(x,{D{\varvec{\textsf {v}}}}):(-q\eta ^{q-1}({\varvec{\textsf {v}}}-{{\varvec{{\mathsf{\lambda }}}}})\otimes D\eta )\, \,dx . \end{aligned}$$

Therefore, due to the coercivity of ${{\mathcal {A}}}$ and the Cauchy–Schwarz inequality we have

$$\begin{aligned} \int _{B_{\sigma r}}G(|{D{\varvec{\textsf {v}}}}|)\eta ^q\,dx\le c\,\int _{B_{\sigma r}}\frac{g(|{D{\varvec{\textsf {v}}}}|)}{|{D{\varvec{\textsf {v}}}}|}\eta ^{q-1}|{D{\varvec{\textsf {v}}}}:(({\varvec{\textsf {v}}}-{{\varvec{{\mathsf{\lambda }}}}})\otimes D\eta )| \,dx=:{\mathcal {K}} \end{aligned}$$

Noting that q is large enough to satisfy $s_G'\ge q'$, we have in turn that $\widetilde{ G}(\eta ^{q-1}t)\le c \eta ^q\widetilde{G}(t)$ and

$$\begin{aligned} \widetilde{G}(\eta ^{q-1}g(t))\le c \eta ^q \widetilde{G}(g(t))\le c \eta ^q G(t). \end{aligned}$$

Then, using Young inequality (20) applied to the integrand of ${\mathcal {K}}$ we get

$$\begin{aligned} \begin{aligned} {\mathcal {K}}&\le \varepsilon \int _{B_{\sigma r}}\widetilde{G}(\eta ^{q-1}|{D{\varvec{\textsf {v}}}}|)\, dx+ c_\varepsilon \int _{B_{\sigma r}}G\left( |{\varvec{\textsf {v}}}-{{\varvec{{\mathsf{\lambda }}}}}|\,|D\eta | /c_1 \right) \, dx\\&\le \varepsilon c\,\int _{B_{\sigma r}} \eta ^{q }G(|{D{\varvec{\textsf {v}}}}| ) \,dx+ c_\varepsilon \int _{B_{\sigma r}} G\left( |{\varvec{\textsf {v}}}-{{\varvec{{\mathsf{\lambda }}}}}|\,|D\eta | /c_1 \right) \, dx \end{aligned} \end{aligned}$$

with arbitrary $\varepsilon <1$. Choosing $\varepsilon$ small enough to absorb the term, and finally by properties of $\eta$ we obtain (29). $\square$

An ${{\mathcal {A}}}$-harmonic function ${\varvec{\textsf {v}}}$ is a minimizer of a functional

$$\begin{aligned} {\varvec{\textsf {v}}}\mapsto \int _{\Omega } G(|{D{\varvec{\textsf {v}}}}|) \, dx. \end{aligned}$$

Therefore, taking the operator independent of the spacial variable

$$\begin{aligned} {{\mathcal {A}}}(x,\xi )=c_a\frac{g(|\xi |)}{|\xi |}\xi \end{aligned}$$

(30)

${{\mathcal {A}}}$-harmonic functions are Lipschitz regular by the following fact.

Lemma 3.12

([35], Lemma 5.8) Suppose G is an N-function of class $C^2((0,\infty ))\cap C([0,\infty ))$, $G,g\in \Delta _2\cap \nabla _2$, ${\varvec{\textsf {w}}}$ is ${{\mathcal {A}}}$-harmonic in $\Omega$ for ${{\mathcal {A}}}$ given by (30). Let $B\subset 2B\Subset \Omega$. Then there exists $c=c({{\texttt {\textit{data}}}})>0$ such that

Proposition 3.13

Suppose a vector field ${{\mathcal {A}}}$ satisfies Assumption (A-vect). If ${\varvec{\textsf {v}}}\in W^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ is ${{\mathcal {A}}}$-harmonic, then for any $\varsigma \in (0,1)$ there exists $R_0=R_0({{\texttt {\textit{data}}}},\varsigma ) \in (0,1]$ such that for any $0< R \le R_0$ and $B_R\Subset \Omega$ it holds that

(31)

whenever $\delta \in (0,1/4],$ where $c_{\textrm{o}}\ge 1$ is a constant depending only on ${{\texttt {\textit{data}}}}$ and $\varsigma$ and $K=K(i_G,s_G)$.

Proof

In this proof, we use a classical perturbation argument, see for instance [50, Theorem 3.8]. It suffices to prove (31) for ${\varvec{\textsf {v}}}$ solving

$$\begin{aligned} -{{\varvec{{\mathsf{\textrm{div}}}}}}\left( a(x)\frac{g(|{D{\varvec{\textsf {v}}}}|)}{|{D{\varvec{\textsf {v}}}}|}{D{\varvec{\textsf {v}}}}\right) =0\quad \text { in }\quad B_1. \end{aligned}$$

Indeed, the general case can be deduced then by considering $\widetilde{{\varvec{\textsf {v}}}}(x)= {{\varvec{\textsf {v}}}(x_0+Rx)}/{R}$ solving $-{{\varvec{{\mathsf{\textrm{div}}}}}}{\bar{{{\mathcal {A}}}}}(x,{D\widetilde{{\varvec{\textsf {v}}}}})=0$ on $B_1(0)$ with

$$\begin{aligned} {\bar{{{\mathcal {A}}}}}(x,\xi )={\bar{a}}(x)\frac{{{\bar{g}}}(|\xi |)}{|\xi |}\xi =\frac{1}{g(R)}{{\mathcal {A}}}(x_0+Rx,R\xi )= {a(x_0+Rx)}\frac{g(R|\xi |)}{g(R)|\xi |}{\xi }. \end{aligned}$$

In this case, the modulus $\omega _{{{\bar{a}}}}$ of continuity of ${{\bar{a}}}$ satisfies $\omega _{{{\bar{a}}}}(r) = \omega _{a}(rR)$.

Note first Propositions 3.10 and 3.11 imply that for any $7/8 \le \sigma ' < \sigma \le 1$ it holds

From the doubling property of G and the upper and lower bound on $\sigma '$ and $\sigma$, we have

Using the triangle inequality and Jensen’s inequality

for $c=c({{\texttt {\textit{data}}}})$. From Lemma 3.9, for any $t \in (0,1/s_{G})$, we have

(32)

Here, in the last line, as $t \mapsto G^{t}(t)$ is a concave function for every $t \in (0, 1/s_{G})$ we have used Jensen’s inequality.

Let ${\varvec{\textsf {w}}}\in {\varvec{\textsf {v}}}+ W^{1,G}_{0}(B_{\sigma }, {{\mathbb {R}}}^m)$ for any $\sigma \in (1,1/2)$ be the weak solution to

$$\begin{aligned} - {{\varvec{{\mathsf{\textrm{div}}}}}}\left( a(x_0) \frac{g(|{D{\varvec{\textsf {w}}}}|)}{|{D{\varvec{\textsf {w}}}}|}{D{\varvec{\textsf {w}}}}\right) = 0 \qquad \text {in } B_{\sigma }. \end{aligned}$$

(33)

Recall that the function ${{\mathcal {V}}}$ describing ellipticity of ${{\mathcal {A}}}$ is defined in (19) and a is a function bounded below by ${c_a}>0$ and with a modulus of continuity $\omega _a$. Testing (33) and (25) against $({\varvec{\textsf {w}}}- {\varvec{\textsf {v}}})$, and applying Lemma 3.2, for any $\epsilon \in (0,1]$ we see

$$\begin{aligned}&\int _{B_{\sigma }} c_a \left| {{\mathcal {V}}}({D{\varvec{\textsf {w}}}})-{{\mathcal {V}}}({D{\varvec{\textsf {v}}}}) \right| ^2 \, dx \\&\quad \le c \int _{B_{\sigma }} a(x_0) \left( \frac{g(|{D{\varvec{\textsf {w}}}}|)}{|{D{\varvec{\textsf {w}}}}|} {D{\varvec{\textsf {w}}}}- \frac{g(|{D{\varvec{\textsf {v}}}}|)}{|{D{\varvec{\textsf {v}}}}|} {D{\varvec{\textsf {v}}}}\right) : ({D{\varvec{\textsf {w}}}}- {D{\varvec{\textsf {v}}}}) \, dx \\&\quad = c \int _{B_{\sigma }} (a(x_0) - a(x)) \frac{g(|{D{\varvec{\textsf {v}}}}|)}{|{D{\varvec{\textsf {v}}}}|} {D{\varvec{\textsf {v}}}}: ({D{\varvec{\textsf {w}}}}-{D{\varvec{\textsf {v}}}}) \, dx \\&\quad \le c \,\omega _{a}(\sigma )\int _{B_{\sigma }} g(|{D{\varvec{\textsf {w}}}}| + |{D{\varvec{\textsf {v}}}}|)\, |{D{\varvec{\textsf {w}}}}-{D{\varvec{\textsf {v}}}}| \, dx \\&\quad \le c\, \epsilon \int _{B_{\sigma }} |{{\mathcal {V}}}({D{\varvec{\textsf {w}}}})-{{\mathcal {V}}}({D{\varvec{\textsf {v}}}})|^2 \, dx + c\,\frac{\omega _{a}(1/2)^2}{\epsilon } \int _{B_{\sigma }} G(|{D{\varvec{\textsf {w}}}}|+|{D{\varvec{\textsf {v}}}}|) \, dx, \end{aligned}$$

where in the last line we used Young’s inequality. As $c=c({{\texttt {\textit{data}}}})$ and $c_a>0$, we can take $\epsilon$ small enough to absorb the first term on the right-hand side. Then Jensen’s inequality implies

$$\begin{aligned} \int _{B_{\sigma }} \left| {{\mathcal {V}}}({D{\varvec{\textsf {w}}}})-{{\mathcal {V}}}({D{\varvec{\textsf {v}}}}) \right| ^2 \, dx&\le c\, \omega _{a}(1/2)^2 \int _{B_{\sigma }} [G(|{D{\varvec{\textsf {v}}}}|) + G(|{D{\varvec{\textsf {w}}}}|)] \, dx. \end{aligned}$$

Since ${\varvec{\textsf {w}}}$ is a minimizer of the integral functional

$$\begin{aligned} {\varvec{\textsf {w}}}\mapsto \int _{B_{\sigma }} G(|{D{\varvec{\textsf {w}}}}|) \, dx, \end{aligned}$$

we have

$$\begin{aligned} \int _{B_{\sigma }} \left| {{\mathcal {V}}}({D{\varvec{\textsf {w}}}})-{{\mathcal {V}}}({D{\varvec{\textsf {v}}}}) \right| ^2 \, dx \le c\, \omega _{a}(1/2)^2 \int _{B_{\sigma }} G(|{D{\varvec{\textsf {v}}}}|) \, dx. \end{aligned}$$

(34)

Since we know the Lipschitz regularity of ${\varvec{\textsf {w}}}$, provided by Lemma 3.12

it follows from (34) that

$$\begin{aligned} \int _{B_{\delta \sigma }} G(|{D{\varvec{\textsf {v}}}}|) \, dx&\le c\, \int _{B_{\delta \sigma }} |{{\mathcal {V}}}({D{\varvec{\textsf {v}}}})|^2 \, dx \\&\le c\, \int _{B_{\delta \sigma }} |{{\mathcal {V}}}({D{\varvec{\textsf {v}}}}) - {{\mathcal {V}}}({D{\varvec{\textsf {w}}}})|^2 \, dx + \int _{B_{\delta \sigma }} |{{\mathcal {V}}}({D{\varvec{\textsf {w}}}})|^2 \, dx \\&\le c\, \int _{B_{\sigma }} |{{\mathcal {V}}}({D{\varvec{\textsf {v}}}}) - {{\mathcal {V}}}({D{\varvec{\textsf {w}}}})|^2 \, dx + \delta ^{n} \int _{B_{\sigma }} |{{\mathcal {V}}}({D{\varvec{\textsf {w}}}})|^2 \, dx \\&\le c \left( \delta ^{n} + \omega _{a}(1/2)^2 \right) \int _{B_{\sigma }} G(|{D{\varvec{\textsf {v}}}}|) \, dx. \end{aligned}$$

We take $R_0=R_{0}({{\texttt {\textit{data}}}},\varsigma )$ small enough to ensure that $\omega _{a}(R_0)^2<\epsilon _{0}$, where $\epsilon _0$ is a constant given in Lemma 3.8. Then Lemma 3.8, Proposition 3.11 and (32) give

(35)

where c depends only on ${{\texttt {\textit{data}}}}$ and $\varsigma$.

Using the Sobolev-Poincaré inequality in $W^{1,1}$ and Jensen’s inequality, we conclude that

what completes the proof. $\square$

4 Measure data ${{\mathcal {A}}}$-harmonic approximation

In this section we provide the tool of crucial meaning for our further reasoning – the approximation of a $W^{1,G}$-function by an ${{\mathcal {A}}}$-harmonic map for weighted operator ${{\mathcal {A}}}$ of an Orlicz growth given by (5). Results in this spirit can be found in [37, 41, 45], but most preeminently for the approximation relevant for application to measure data problems we refer to [65, Theorem 4.1].

We define an auxiliary function

$$\begin{aligned} H_s(t)=\int _{0}^{t} \frac{g(r)^{1-s} G(r)^{s}}{r} \, dr\qquad \text {for }\ s \in [0,1/2). \end{aligned}$$

(36)

It is readily checked that when $s>\max \{2-i_{G},0\}$, $H_s$ is a Young function satisfying

$$\begin{aligned} H_s(t) \approx g(t)^{1-s} G(t)^{s} \end{aligned}$$

(37)

with intrinsic constants depending only on $i_G,s_G$ and s. In fact, $H_s\in \Delta _2\cap \nabla _2$ since

$$\begin{aligned} 0 < s + i_G -2\le \frac{t H_s''(t)}{H_s'(t)} = \frac{(1-s) t g'(t)}{g(t)} + \frac{s t g(t)}{G(t)} \le s + s_G -2. \end{aligned}$$

(38)

Furthermore, there exist $\epsilon ,c,t_0>0$, such that $H_s(t)\ge c t^{1+\epsilon }$ and $H_s(t)\ge c g^{1+\epsilon }(t)$ for all $t\ge t_0.$

Theorem 4.1

Under Assumption (A-vect) let $\varepsilon >0$, $\gamma \in (0,1/(s_Gn))$, and

$$\begin{aligned} \max \{2-i_{G},0\}< s <s_{\textrm{m}}:=\frac{i_G-\gamma s_G n}{i_G+ s_Gn}. \end{aligned}$$

(39)

Suppose that ${\varvec{\textsf {u}}}\in W^{1,G}(B_r(x_0),{{\mathbb {R}}^{m}})$ satisfies

(40)

then there exists $\delta =\delta ({{\texttt {\textit{data}}}},s,M,\varepsilon )\in (0,1]$ such that if ${\varvec{\textsf {u}}}$ is almost ${{\mathcal {A}}}$-harmonic in a sense that for every ${{\varvec{{\mathsf{\varphi }}}}}\in W^{1,G}_0(B_r(x_0),{{\mathbb {R}}^{m}})\cap L^\infty (B_r(x_0),{{\mathbb {R}}^{m}})$ it holds

(41)

then there exists an ${{\mathcal {A}}}$-harmonic map ${\varvec{\textsf {v}}}\in W^{1,G}(B_{r/2}(x_0),{{\mathbb {R}}^{m}})$ satisfying

(42)

together with

(43)

where $c=c({{\texttt {\textit{data}}}})>0.$

Remark 4.2

The limitation that G has to be superquadratic ($i_G\ge 2$) can be a little bit relaxed in Theorem 4.1 and later on the restriction is not needed. The key property is to ensure that the range of admissible s from (39) is nonempty. We need to assume that $i_G$ is either bigger or equal to 2, or close to 2 in a sense that

$$\begin{aligned} 2-i_G <\frac{i_G-1}{i_G+ s_Gn}. \end{aligned}$$

Proof

The plan is to first establish suitable a priori estimates for the rescaled problem and then proceed with the proof via contradiction. The proof is presented in 6 steps.

Step 1. Scaling. We fix arbitrary ${{\varvec{{\mathsf{\varphi }}}}}\in W^{1,G}_0(B_r(x_0),{{\mathbb {R}}^{m}})\cap L^\infty (B_r(x_0),{{\mathbb {R}}^{m}})$ satisfying (41). Let us change variables putting

$$\begin{aligned} {\bar{\varvec{\textsf {u}}}}(x):=\frac{{\varvec{\textsf {u}}}(x_0+rx)}{Mr},\ {\bar{{{\mathcal {A}}}}}(x_0+rx,{{\varvec{{\mathsf{\xi }}}}})={{\mathcal {A}}}(x_0+rx,M{{\varvec{{\mathsf{\xi }}}}}),\ \text {and}\ {{\varvec{{\mathsf{\eta }}}}}(x):=\frac{{{\varvec{{\mathsf{\varphi }}}}}(x_0+rx)}{r}.\nonumber \\ \end{aligned}$$

(44)

Then ${\bar{{{\mathcal {A}}}}}$ satisfies the same conditions as ${{\mathcal {A}}}$ with the functions ${{\bar{g}}}(t):=g(Mt)$ and ${{\bar{G}}}(t):=G(Mt)/M$ (with ${{\bar{G}}}'={{\bar{g}}}$), and the constants depending on ${{\texttt {\textit{data}}}}.$ Of course in such a case $i_G=i_{{\bar{G}}}$ and $s_G=s_{{\bar{G}}}.$

Having (40) and (41), by denoting the unit ball by $B_1$, we get further that

(45)

(46)

Step 2. A priori estimates. We choose $q\ge s_G$, pick

$$\begin{aligned} {{\varvec{{\mathsf{\eta }}}}}:=\phi ^qT_k({\bar{\varvec{\textsf {u}}}})\quad \text {with some}\ \phi \in C_c^\infty (B_1),\ 0\le \phi \le 1,\ k\ge 0, \end{aligned}$$

(47)

and denote

$$\begin{aligned} {{\textsf{P}}}:=\frac{{\bar{\varvec{\textsf {u}}}}\otimes {\bar{\varvec{\textsf {u}}}}}{|{\bar{\varvec{\textsf {u}}}}|^2}. \end{aligned}$$

Then

$$\begin{aligned} {D{{\varvec{{\mathsf{\eta }}}}}}&=\mathbb {1}_{\{|{\bar{\varvec{\textsf {u}}}}|\le k\}}(\phi ^q {D{\bar{\varvec{\textsf {u}}}}}+q\phi ^{q-1}{\bar{\varvec{\textsf {u}}}}\otimes D\phi )\\&\quad +\mathbb {1}_{\{|{\bar{\varvec{\textsf {u}}}}|> k\}}(\phi ^q ({{\textsf{Id}}}-{{{\textsf{P}}}}) {D{\bar{\varvec{\textsf {u}}}}}+q\phi ^{q-1}{\bar{\varvec{\textsf {u}}}}\otimes D\phi ). \end{aligned}$$

We use (47) in (46) to get

$$\begin{aligned}&\left| \int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|\le k\}} {\bar{{{\mathcal {A}}}}}(x,{D{\bar{\varvec{\textsf {u}}}}}):(\phi ^q {D{\bar{\varvec{\textsf {u}}}}}+q\phi ^{q-1}{\bar{\varvec{\textsf {u}}}}\otimes D\phi )\,dx\right. \nonumber \\&\quad \left. +\int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|> k\}} {\bar{{{\mathcal {A}}}}}(x,{D{\bar{\varvec{\textsf {u}}}}}):(\phi ^q ({\textsf{Id}}-{{\textsf{P}}}) {D{\bar{\varvec{\textsf {u}}}}}+q\phi ^{q-1}{\bar{\varvec{\textsf {u}}}}\otimes D\phi )\,dx\right| \le {\delta }|B_1|\Vert {{\varvec{{\mathsf{\eta }}}}}\Vert _{L^\infty (B_1,{{\mathbb {R}}^{m}})}. \end{aligned}$$

(48)

Since ${\bar{{{\mathcal {A}}}}}$ has the quasi-diagonal structure resulting from (5) and

$$\begin{aligned} {D{\bar{\varvec{\textsf {u}}}}}:\big (({\textsf{Id}}-{{\textsf{P}}}){D{\bar{\varvec{\textsf {u}}}}}\big )=|{D{\bar{\varvec{\textsf {u}}}}}|^2-\frac{D_j{\bar{\varvec{\textsf {u}}}}^\alpha {\bar{\varvec{\textsf {u}}}}^\alpha D_j{\bar{\varvec{\textsf {u}}}}^\beta {\bar{\varvec{\textsf {u}}}}^\beta }{|{\bar{\varvec{\textsf {u}}}}|^2}=|{D{\bar{\varvec{\textsf {u}}}}}|^2-\frac{\sum _{j=1}^m\langle D_j {\bar{\varvec{\textsf {u}}}},{\bar{\varvec{\textsf {u}}}}\rangle ^2}{|{\bar{\varvec{\textsf {u}}}}|^2}\ge 0, \end{aligned}$$

we infer that

$$\begin{aligned} {\bar{{{\mathcal {A}}}}}(x, {D{\bar{\varvec{\textsf {u}}}}}):\big (({\textsf{Id}}-{{\textsf{P}}}){D{\bar{\varvec{\textsf {u}}}}}\big )\ge 0. \end{aligned}$$

(49)

By rearranging terms in (48), applying Lemma 3.1, noting that $\Vert {{\varvec{{\mathsf{\eta }}}}}\Vert _{L^\infty (B_1,{{\mathbb {R}}^{m}})}\le k$, and dropping a nonnegative term due to (49), we get for some $c=c({{\texttt {\textit{data}}}},q)$

$$\begin{aligned}&\int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|\le k\}}{{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|) \phi ^q\,dx\le c \int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|\le k\}} \frac{{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}{|{D{\bar{\varvec{\textsf {u}}}}}|}\phi ^{q-1}\big |{D{\bar{\varvec{\textsf {u}}}}}:({\bar{\varvec{\textsf {u}}}}\otimes D\phi )\big |\,dx\\&\quad +\,c\int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|> k\}} \frac{k}{|{\bar{\varvec{\textsf {u}}}}|} \frac{{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}{|{D{\bar{\varvec{\textsf {u}}}}}|}\phi ^{q-1}\big |{D{\bar{\varvec{\textsf {u}}}}}:({\bar{\varvec{\textsf {u}}}}\otimes D\phi )\big |\,dx+ c|B_1|\delta k. \end{aligned}$$

We estimate the first term on the right-hand side of the last display by the use of Young inequality with a parameter, use Lemma 3.1, and we absorb one term. The second term can be estimated by the Cauchy–Schwarz inequality. Altogether we obtain

$$\begin{aligned} \int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|< k\}}{{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|) \phi ^q\,dx&\le c \int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|< k\}}{{\bar{G}}}(|{\bar{\varvec{\textsf {u}}}}|\,|D\phi |) \,dx+ c|B_1|\delta k\nonumber \\&\quad +\,ck \int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|\ge k\}} {{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}\phi ^{q-1}| D\phi |\,dx \end{aligned}$$

(50)

for some $c=c({{\texttt {\textit{data}}}},q)$.

Step 3. Summability of ${\bar{\varvec{\textsf {u}}}}$ and $D{\bar{\varvec{\textsf {u}}}}$. We choose $k=t$ in (50), then multiply this line by $(1+t)^{-(\gamma +2)},$ where $\gamma >0,$ integrate it from zero to infinity and apply Cavalieri’s principle (Lemma 3.6) twice (with $\nu _1={{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|) \phi ^q$ and $\nu _2={{\bar{G}}}(|{\bar{\varvec{\textsf {u}}}}|\,|D\phi |)$. Altogether we get

$$\begin{aligned} \frac{1}{1+\gamma } \int _{B_1 }\frac{{{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|) \phi ^q}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\,dx&\le \frac{c}{1+\gamma } \int _{B_1 }\frac{{{\bar{G}}}(|{\bar{\varvec{\textsf {u}}}}|\,|D\phi |) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\,dx+ \frac{c}{\gamma }\delta \nonumber \\&\quad +\, c\,\int _0^\infty \frac{t}{(1+t)^{\gamma +2}}\int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|> t\}} {{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}\phi ^{q-1}| D\phi |\,dx\,dt. \end{aligned}$$

(51)

The right-most term in the last display can be estimated as follows

$$\begin{aligned}&c\,\int _0^\infty \frac{t}{(1+t)^{\gamma +2}}\int _{B_1\cap \{|{\bar{\varvec{\textsf {u}}}}|> t\}} {{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}\phi ^{q-1}| D\phi |\,dx\,dt\nonumber \\&\quad \le c \int _0^\infty \frac{1}{(1+t)^{\gamma +1}}dt\,\int _{B_1 } {{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}\phi ^{q-1}|D\phi |\,dx \nonumber \\&\quad \le \frac{c}{\gamma } \int _{B_1 } {{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}\phi ^{q-1}| D\phi |\,dx . \end{aligned}$$

(52)

To estimate it further we note that q is large enough to satisfy $s_G'\ge q'$, there exist $c_0,c_1>0$ depending on $i_G,s_G$, such that we have ${{{\bar{G}}}}^*(c_0\phi ^{q-1}{{\bar{g}}}(t))\le c_1 \phi ^q {{{\bar{G}}}}^*({{\bar{g}}}(t))\le \phi ^q {{\bar{G}}}(t).$ Then, using Young inequality (20) applied to the integrand in (52) and the above observation we get

$$\begin{aligned} \begin{aligned}&\int _{B_1 } {{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}\phi ^{q-1}|D\phi |\,dx\\&\quad \le \frac{1}{2(1+\gamma )} \int _{B_{1}}\frac{{{{\bar{G}}}^*}\left( {c_0}{\phi ^{q-1}{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|) }\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} \, dx + \widetilde{c} \int _{B_{1}}\frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\,dx \\&\quad \le \frac{1}{2(1+\gamma )} \int _{B_{1}}\frac{{{\bar{G}}}\left( |{D{\bar{\varvec{\textsf {u}}}}}| \right) \phi ^{q }}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} \,dx + \widetilde{c} \int _{B_{1}}\frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\, dx \end{aligned} \end{aligned}$$

with $\widetilde{c}=\widetilde{c}(\gamma ,i_G,s_G)$. By applying this estimate in (51) and rearranging terms we obtain

$$\begin{aligned} \int _{B_1 }\frac{{{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|) \phi ^q}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\,dx&\le {c} \int _{B_{1}}\frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\, dx + c\,\delta \frac{1+\gamma }{\gamma } . \end{aligned}$$

(53)

Observe that $1/(s_G n)< i_G-1$, as otherwise the condition required by Remark 4.2 is violated. Recall that since $\gamma <1/(s_G n)$, we have $\gamma < i_G-1$. Let us set

$$\begin{aligned} \vartheta (x):=\frac{{{\bar{G}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} \end{aligned}$$

(54)

and notice that since $|{D|{\bar{\varvec{\textsf {u}}}}|}|\le |{D{\bar{\varvec{\textsf {u}}}}}|$, using Lemma 3.1, we can estimate

$$\begin{aligned} \left| {D\vartheta }\right|&=\frac{\left| {D( {{\bar{G}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)}(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }-{D\left( (1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }\right) }{{\bar{G}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)\right| }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{2+2\gamma }}\nonumber \\&\le \, \frac{{{\bar{g}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)\left| {D((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)}\right| }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} + {(1+\gamma )} \frac{\left| D{{\bar{\varvec{\textsf {u}}}}} \right| {{\bar{G}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{2+\gamma }}\nonumber \\&\le \, \frac{{{\bar{g}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)|{D{\bar{\varvec{\textsf {u}}}}}|\phi ^q}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\nonumber \\&\quad + q \, \frac{{{\bar{g}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)(1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^{q-1}\left| {D\phi }\right| }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} + {(1+\gamma )} \frac{ \left| {D {\bar{\varvec{\textsf {u}}}}}\right| }{1+|{\bar{\varvec{\textsf {u}}}}|}\vartheta \nonumber \\&{\le c(\gamma )\frac{ \left| {D {\bar{\varvec{\textsf {u}}}}}\right| }{1+|{\bar{\varvec{\textsf {u}}}}|}\vartheta +c \, \frac{\left| {D\phi }\right| }{\phi }\vartheta .} \end{aligned}$$

(55)

For the later use in Step 6, we emphasize the dependence of constants on $\gamma$ by denoting $c(\gamma )$. Note that every $c(\gamma )$ in (55)-(59) is an increasing function of $\gamma$.

Since $q \ge s_G$, for any $\kappa \in [1,i_{G})$ we see

$$\begin{aligned} \left| {D\left( \vartheta ^{\frac{1}{\kappa }}\right) }\right| \le \frac{1}{\kappa } \vartheta ^{\frac{1}{\kappa }-1} \left| {D\vartheta }\right| \le {c(\gamma )} \, \vartheta ^{\frac{1}{\kappa }} \frac{\left| {D{\bar{\varvec{\textsf {u}}}}}\right| }{1+|{\bar{\varvec{\textsf {u}}}}|} + {c }\, \vartheta ^{\frac{1}{\kappa }} \frac{\left| {D \phi } \right| }{\phi } \end{aligned}$$

and

$$\begin{aligned} \left| {D\left( \vartheta ^{\frac{1}{\kappa }}\right) }\right| ^{\kappa }&\le c(\gamma ) \frac{G((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^{q})}{\left[ (1+|{\bar{\varvec{\textsf {u}}}}|) \phi ^{q} \right] ^{\kappa }} \frac{\left( |{D{\bar{\varvec{\textsf {u}}}}}| \phi ^{q} \right) ^{\kappa }}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} + c \frac{G(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} |{D \phi }|^{\kappa }. \end{aligned}$$

(56)

To proceed further, we define an auxiliary function $h_\kappa$ by setting

$$\begin{aligned} h_{\kappa }^{-1}(t):= \int _{0}^{t} \frac{1}{\left[ G^{-1}(\tau )\right] ^{\kappa }} \, d\tau . \end{aligned}$$

A straightforward calculation gives

$$\begin{aligned} -1< - \frac{\kappa }{i_G} \le \frac{t [h_{\kappa }^{-1}]''(t)}{[h_{\kappa }^{-1}]'(t)} = - \frac{\kappa t [G^{-1}]'(t)}{G^{-1}(t)} \le - \frac{\kappa }{s_G} <0, \end{aligned}$$

(57)

which implies that $h_{\kappa }^{-1}$ is an increasing concave function on $[0, \infty )$. Note that (57) also gives

$$\begin{aligned} h_{\kappa }\left( \frac{G(t)}{t^{\kappa }} \right) \approx G(t) \end{aligned}$$

with intrinsic constants depending on $i_G,s_G$ and $\kappa$ only. Moreover, we have

$$\begin{aligned} {{\bar{g}}}(t)[h_{\kappa }^{-1}]'({{\bar{G}}}(t))=[h_{\kappa }^{-1}({{\bar{G}}}(t))]' = \frac{d}{dt} \left( \int _{0}^{G(t)} \frac{1}{[G^{-1}(\tau )]^{\kappa }} \, d \tau \right) = \frac{{{\bar{g}}}(t)}{t^{\kappa }}, \end{aligned}$$

and so

$$\begin{aligned}{}[\widetilde{h_{\kappa }}]^{-1}(G(t)) \approx h_{\kappa }'(h_{\kappa }^{-1}({{\bar{G}}}(t)) = \frac{1}{[h_{\kappa }^{-1}]'(h_{\kappa }(h_{\kappa }^{-1}(G(t))))}= \frac{1}{[h_{\kappa }^{-1}]'(G(t))} = t^{\kappa }. \end{aligned}$$

Hence, $\widetilde{h_{\kappa }}(t) \approx {{\bar{G}}}(t^{1/\kappa })$.

Applying Young’s inequality (20) with the pair of Young functions $(h_{\kappa },\widetilde{h_{\kappa }})$ to (56), for any $\epsilon _0 \in (0,1)$ we discover

$$\begin{aligned} \left| {D\left( \vartheta ^{\frac{1}{\kappa }}\right) }\right| ^{\kappa }&\le \epsilon _{0} \frac{G((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^{q})}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} + c(\epsilon _0)c(\gamma ) \frac{{{\bar{G}}}\left( |{D{\bar{\varvec{\textsf {u}}}}}| \phi ^{q} \right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} + c \frac{G(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} |{D \phi }|^{\kappa }. \end{aligned}$$

By the classical Sobolev inequality we get that

$$\begin{aligned} \left( \int _{B_1} |\vartheta |^\frac{n}{n-\kappa } \,dx\right) ^\frac{n-\kappa }{n}\le c \int _{B_1} \left| {D \left( \vartheta ^{\frac{1}{\kappa }}\right) }\right| ^{\kappa } \,dx. \end{aligned}$$

(58)

Merging (51), (53), (54), (55) and (58) and taking $\epsilon _0$ small enough, we get

$$\begin{aligned} \left( \int _{B_1}\left( \frac{{{\bar{G}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^\frac{n}{n-\kappa }\,dx\right) ^\frac{n-\kappa }{n}\le {{\bar{c}}} \left( \int _{B_{1}}\frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\, dx + \frac{1}{\gamma }\right) . \end{aligned}$$

(59)

It is worth to mention that ${{\bar{c}}}={\bar{c}}({{\texttt {\textit{data}}}},\gamma ,\delta )>0$ depends on ${{\texttt {\textit{data}}}}, \gamma$ and $\delta$ and it is an increasing function of $\gamma$ and $\delta$. Since $\gamma \in (0,1/(s_G n))$ is fixed, we may choose $\alpha$ such that

$$\begin{aligned} \alpha \in \left( 1,\frac{n}{n-\kappa }\right) \qquad \text {and} \qquad \frac{\alpha s_G\gamma }{\alpha -1} \le 1. \end{aligned}$$

(60)

Then

$$\begin{aligned} \int _{B_{1}}(1+|{\bar{\varvec{\textsf {u}}}}|)^{\frac{\alpha s_G\gamma }{\alpha -1}} \, dx\le \int _{B_{1}} 1+|{\bar{\varvec{\textsf {u}}}}|\,dx\le 1+|B_1| \end{aligned}$$

and we can estimate

$$\begin{aligned}&\int _{B_{1}}\frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}})|)^{1+\gamma }|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\, dx \le \int _{B_{1}}\frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma -\gamma s_G}}\, dx \nonumber \\&\quad \le \left( \int _{B_{1}}(1+|{\bar{\varvec{\textsf {u}}}}|)^{\frac{\alpha s_G\gamma }{\alpha -1}} \, dx\right) ^\frac{\alpha -1}{\alpha }\left( \int _{B_{1}} \left( \frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^{\alpha } \, dx\right) ^\frac{1}{\alpha }\nonumber \\&\quad \le c\left( \int _{B_{1}} \left( \frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^{\alpha } \, dx\right) ^\frac{1}{\alpha }. \end{aligned}$$

(61)

Thus, from (59) and (61) we obtain

$$\begin{aligned} \left( \int _{B_1}\left( \frac{{{\bar{G}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^\frac{n}{n-\kappa }\,dx\right) ^\frac{n-\kappa }{n}\le c\left( \int _{B_{1}}\left( \frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^{\alpha }\, dx\right) ^\frac{1}{\alpha } + c .\qquad \end{aligned}$$

(62)

For $7/8 \le r_1 < r_2\le 1$, we take a cut-off function $\phi$ satisfying

$$\begin{aligned} \phi \equiv 1\quad \text {on }\ B_{r_1}\qquad \text {and}\qquad |D\phi |\le \frac{100}{r_2-r_1}. \end{aligned}$$

It then follows from the doubling property of ${{\bar{G}}}$ and Lemma 3.9 that for any $\upsilon \in (0,1/s_G)$ we have

$$\begin{aligned} \left( \int _{B_{7/8}}\left( \frac{{{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^\frac{n}{n-\kappa }\,dx\right) ^\frac{n-\kappa }{n}&\le c\left( \int _{B_{1}}\left( \frac{{{\bar{G}}}\left( 1+|{\bar{\varvec{\textsf {u}}}}|\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^{\upsilon }\, dx\right) ^\frac{1}{\upsilon } + c \\&\le c \left( \int _{B_{1}} ({{\bar{G}}}\left( 1+|{\bar{\varvec{\textsf {u}}}}|\right) )^{\upsilon }\, dx\right) ^\frac{1}{\upsilon } + c\\&\le {{\bar{G}}}\left( \int _{B_{1}} (1+|{\bar{\varvec{\textsf {u}}}}|)\, dx \right) + c. \end{aligned}$$

In the last line, we have used Jensen’s inequality with the concave function $t \mapsto {{\bar{G}}}(t)^{\upsilon }$. Recalling (45), we obtain

$$\begin{aligned} \left( \int _{B_{7/8}}\left( \frac{{{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^\frac{n}{n-\kappa }\,dx\right) ^\frac{n-\kappa }{n} \le c \end{aligned}$$

(63)

for some $c=c({{\texttt {\textit{data}}}},\gamma )>0.$

To proceed further, we recall $H_s$ defined in (36) with s from (39). By (37) function $H_s$ satisfies also $H_s(Mt) \approx _{{{\texttt {\textit{data}}}}} {{\bar{g}}}(t)^{1-s} {{\bar{G}}}(t)^{s}$. We apply Young’s inequality, (53), and (63) with suitable choice of $\phi$. In turn we see

$$\begin{aligned}&\int _{B_{\rho _1}} H_s(M|{D{\bar{\varvec{\textsf {u}}}}}|) \,dx \lesssim _{{{\texttt {\textit{data}}}}} \int _{B_{ \rho _1}} \frac{{{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)^{1-s}{{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)^{s}}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }\,dx \nonumber \\&\quad \le \int _{B_{ \rho _1}} \frac{{{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\,dx + \int _{B_{ \rho _1}} \frac{ {{\bar{g}}}(|{D{\bar{\varvec{\textsf {u}}}}}|) (1+|{\bar{\varvec{\textsf {u}}}}|)^{\frac{1+\gamma }{1-s}}}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} \,dx \nonumber \\&\quad \le \int _{B_{ \rho _1}} \frac{{{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\,dx + \int _{B_{ \rho _1}} \frac{ {{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|) (1+|{\bar{\varvec{\textsf {u}}}}|)^{\frac{(s + \gamma ) s_G}{(1-s)}}}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }} \,dx \nonumber \\&\quad \le \int _{B_{ \rho _1}} \frac{{{\bar{G}}}(|{D{\bar{\varvec{\textsf {u}}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\,dx\nonumber \\&\qquad + \left( \int _{B_{ \rho _1}}\left( \frac{{{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{1+\gamma }}\right) ^\frac{n}{n-\kappa }\,dx\right) ^\frac{n-\kappa }{n} \left( \int _{B_{ \rho _1}} (1+|{\bar{\varvec{\textsf {u}}}}|)^\frac{(s + \gamma ) s_G n}{(1-s)\kappa }\,dx \right) ^\frac{\kappa }{n} \end{aligned}$$

(64)

In order to use (45) we need to have

$$\begin{aligned} \frac{(s + \gamma ) s_G n}{(1-s)\kappa } \le 1. \end{aligned}$$

(65)

Observe that by Remark 4.2 we have $2-i_G <({i_G-\gamma s_G n})/({i_G+ s_Gn}),$ and we can choose $\kappa \in [1, i_G)$ such that

$$\begin{aligned} \max \{2-i_{G},0\}< s \le {\frac{\kappa -\gamma s_G n}{\kappa + s_Gn}}<s_{\textrm{m}} \end{aligned}$$

and the bound (65) follows. Then (64) combined with (53), (61) and (63) implies for $s<s_{\textrm{m}}$ that

$$\begin{aligned} \int _{B_{3/4}}H_s(M|{D{\bar{\varvec{\textsf {u}}}}}|)\,dx\le c_{\textrm{ap}}=c_{\textrm{ap}}({{\texttt {\textit{data}}}},\gamma ). \end{aligned}$$

(66)

Step 4. Contradiction argument. We state the counter–assumption.

Namely, we assume that there exists $\varepsilon$ and sequences of balls $\{B_{r_j}(x_j)\}$ and almost ${{\mathcal {A}}}$-harmonic maps $\{{\varvec{\textsf {u}}}_j\}\subset W^{1,G}(B_{r_j}(x_j),{{\mathbb {R}}^{m}})$ such that

(67)

(68)

for all ${{\varvec{{\mathsf{\varphi }}}}}\in W^{1,G}_0(B_{r_j}(x_j),{{\mathbb {R}}^{m}})\cap L^\infty (B_{r_j}(x_j),{{\mathbb {R}}^{m}}),$ but such that

(69)

whenever ${\varvec{\textsf {v}}}\in W^{1,G}(B_{r/2}(x_0),{{\mathbb {R}}^{m}})$ is an ${{\mathcal {A}}}$-harmonic map in $B_{r/2}(x_0)$ satisfying

(70)

where $c=c({{\texttt {\textit{data}}}})>0.$

Let ${\bar{\varvec{\textsf {u}}}}$ be scaled as in (44), but with the use of $x_j$ and $r_j,$ that is we set

$$\begin{aligned} {\bar{\varvec{\textsf {u}}}}_j(x):=\frac{{\varvec{\textsf {u}}}(x_j+r_jx)}{Mr_j}\quad \text {and}\quad {\bar{{{\mathcal {A}}}}}(x_j+r_jx,{{\varvec{{\mathsf{\xi }}}}}):={{\mathcal {A}}}(x_j+r_jx,M{{\varvec{{\mathsf{\xi }}}}}). \end{aligned}$$

In such a case by (67) we get that

(71)

so by (68) we infer that for all ${{\varvec{{\mathsf{\eta }}}}}={{\varvec{{\mathsf{\varphi }}}}}(x_j+r_jx)/r_j\in W^{1,G}_0(B_1,{{\mathbb {R}}^{m}})\cap L^\infty (B_1,{{\mathbb {R}}^{m}})$ it holds

(72)

and

(73)

whenever ${\bar{\varvec{\textsf {{v}}}}}\in W^{1,G}(B_{1/2}(x_0),{{\mathbb {R}}^{m}})$ is an ${\bar{{{\mathcal {A}}}}}$-harmonic map in $B_{1/2}(x_0)$ satisfying

(74)

where $c=c({{\texttt {\textit{data}}}})>0.$

Since ${\bar{\varvec{\textsf {u}}}}$ satisfies (71), we have (66) for $s_{\textrm{o}}<s_{\textrm{m}}$ from (39). Therefore

$$\begin{aligned} \int _{B_{3/4}}H_{s_{\textrm{o}}}(M|{D{\bar{\varvec{\textsf {u}}}}}_j|)\,dx\le C\quad \text {for}\quad C=C({{\texttt {\textit{data}}}},\gamma ). \end{aligned}$$

We fix any $s<s_{\textrm{o}}$ from the range (39). Then we pick $\epsilon >0$ for which there exist $c,t_0>0$, such that $H_s(t)\ge c t^{1+\epsilon }$ and $H_s(t)\ge c g^{1+\epsilon }(t)$ for all $t\ge t_0.$ In turn, we conclude with the following estimates uniform in j

$$\begin{aligned} \int _{B_{3/4}}g^{1+\epsilon }(M|{{D{\bar{\varvec{\textsf {u}}}}}_j}|)\,dx\le c_1\quad \text {and}\quad \int _{B_{3/4}}(M|{{D{\bar{\varvec{\textsf {u}}}}}_j}|)^{1+\epsilon }\,dx\le c_2 \end{aligned}$$

(75)

with $c_1,c_2$ depending on ${{\texttt {\textit{data}}}}$ and $\gamma$ only. Further we infer that there exist

$$\begin{aligned} {{\widetilde{\varvec{\textsf {u}}}}}\in W^{1,H_s}(B_{3/4},{{\mathbb {R}}^{m}}),\quad {{{\mathfrak {A}}}}\in L^{1+\epsilon }(B_{3/4},{{\mathbb {R}}^{n\times m}}),\quad \,\text {and}\, \quad {\mathfrak {h}}\in L^{H_s}(B_{3/4}) \end{aligned}$$

such that up to a subsequence

$$\begin{aligned} \begin{aligned}&{{D{\bar{\varvec{\textsf {u}}}}}_j}- {{D{{\widetilde{\varvec{\textsf {u}}}}}}}\rightharpoonup 0\qquad \text {in } \ L^{H_s}(B_{3/4},{{\mathbb {R}}^{n\times m}}),\\&|{{D{\bar{\varvec{\textsf {u}}}}}_j}-{{D{{\widetilde{\varvec{\textsf {u}}}}}}}| \rightharpoonup {\mathfrak {h}}\qquad \text {in } \ L^{H_s}(B_{3/4}),\\&{\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j}) \rightharpoonup {{{\mathfrak {A}}}}\qquad \text {in } \ L^{1+\epsilon }(B_{3/4},{{\mathbb {R}}^{n\times m}}),\\&{\bar{\varvec{\textsf {u}}}}_j\rightarrow {{\widetilde{\varvec{\textsf {u}}}}}\quad \text { strongly in }\, \ L^{H_s}(B_{3/4},{{\mathbb {R}}^{m}})\ \,\text { and a.e. in }\,B_{3/4}. \end{aligned} \end{aligned}$$

(76)

By (71), lower semicontinuity of a functional , and (66) we have

(77)

Step 5. Strong convergence of gradients. Our aim is now to prove that

$$\begin{aligned} {{D{\bar{\varvec{\textsf {u}}}}}_j}\rightarrow {{D{{\widetilde{\varvec{\textsf {u}}}}}}}\qquad \text {in }\ L^{H_s}(B_{3/4},{{\mathbb {R}}^{n\times m}}). \end{aligned}$$

(78)

For this we need to show that ${\mathfrak {h}}\in L^{H_s}(B_{3/4})$ from (76) satisfies

$$\begin{aligned} {\mathfrak {h}}=0 \end{aligned}$$

(79)

a.e. in $B_{3/4}$. This almost everywhere and weak convergence in $L^1$ implies strong $L^1$-convergence of ${{D{\bar{\varvec{\textsf {u}}}}}_j}\rightarrow {{D{{\widetilde{\varvec{\textsf {u}}}}}}}$ in $B_{3/4}$. Using the monotonicity property of $H_s$ and (74), for sufficiently small $\widetilde{\epsilon }\in (0, \frac{1}{s_{H_{s}}})$ we have

$$\begin{aligned}&\int _{B_{3/4}} H_s ( M |{{D{\bar{\varvec{\textsf {u}}}}}_j} - {{D{{\widetilde{\varvec{\textsf {u}}}}}}}|) \,dx \nonumber \\&\quad \le \int _{B_{3/4}} H_s( M |{{D{\bar{\varvec{\textsf {u}}}}}_j} - {{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)^{\widetilde{\epsilon }^2} H_s( M |{{D{\bar{\varvec{\textsf {u}}}}}_j}| + M |{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)^{1-\widetilde{\epsilon }^2} \,dx \nonumber \\&\quad \le \bigg ( \int _{B_{3/4}} H_s( M |{{D{\bar{\varvec{\textsf {u}}}}}_j} - {{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)^{\widetilde{\epsilon }} \,dx \bigg )^{\widetilde{\epsilon }} \bigg ( \int _{B_{3/4}} H_s( M |{{D{\bar{\varvec{\textsf {u}}}}}_j}| + M |{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)^{1 + \widetilde{\epsilon }} \,dx \bigg )^{1-\widetilde{\epsilon }} \nonumber \\&\quad \le c \bigg ( \int _{B_{3/4}} H_s( M |{{D{\bar{\varvec{\textsf {u}}}}}_j} - {{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)^{\widetilde{\epsilon }} \,dx \bigg )^{\widetilde{\epsilon }}. \end{aligned}$$

(80)

Denoting

$$\begin{aligned} \Psi (t) = \int _{0}^{t} \frac{H_{s}^{-1}(\tau ^{1/\widetilde{\epsilon }})}{\tau } \, d \tau , \end{aligned}$$

one can immediately check

$$\begin{aligned} \frac{t \Psi ''(t)}{\Psi '(t)} = \frac{t^{1/\widetilde{\epsilon }}}{\widetilde{\epsilon }H_{s}' ( H_{s}^{-1}(t^{1/\widetilde{\epsilon }}) ) H_{s}^{-1}(t^{1/\widetilde{\epsilon }}) } - 1 \ge \frac{1}{\widetilde{\epsilon }s_{H_{s}}} -1 >0, \end{aligned}$$

and so $\Psi$ is a Young function. We then apply Jensen’s inequality to (80) to obtain

$$\begin{aligned} \int _{B_{3/4}} H_s ( M |{{D{\bar{\varvec{\textsf {u}}}}}_j} - {{D{{\widetilde{\varvec{\textsf {u}}}}}}}|) \,dx \le c \bigg [ H_{s} \bigg ( M \int _{B_{3/4}} |{{D{\bar{\varvec{\textsf {u}}}}}_j} - {{D{{\widetilde{\varvec{\textsf {u}}}}}}}| \,dx \bigg ) \bigg ]^{\widetilde{\epsilon }^2} {\mathop {\longrightarrow }\limits ^{j \rightarrow \infty }} 0. \end{aligned}$$

Hence, it remains to show (79) to obtain (78).

We pick ${\bar{x}}$ being a Lebesgue’s point simultaneously for ${{{\widetilde{\varvec{\textsf {u}}}}}}, {{D{{\widetilde{\varvec{\textsf {u}}}}}}}, {\mathfrak {h}},{{{\mathfrak {A}}}}$, that is

(81)

and

$$\begin{aligned} |{{{\widetilde{\varvec{\textsf {u}}}}}}({\bar{x}})|+|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}({\bar{x}})|+|{\mathfrak {h}}({\bar{x}})|+|{{{\mathfrak {A}}}}({\bar{x}})|<\infty . \end{aligned}$$

(82)

Almost every point of $B_{3/4}$ satisfies this conditions. Thus it is enough to show that (79) holds for ${\bar{x}}$.

We restrict our attention to $\varrho$ small enough for $B_\varrho ({{\bar{x}}})\subset B_{3/4}$ and we set the linearization of ${{{\widetilde{\varvec{\textsf {u}}}}}}$ at ${\bar{x}}$

$$\begin{aligned} {{\varvec{{\mathsf{\ell _\varrho }}}}}(x):=({{{\widetilde{\varvec{\textsf {u}}}}}})_{B_{\varrho }({{\bar{x}}})} + {{D{{\widetilde{\varvec{\textsf {u}}}}}}}({\bar{x}}):(x-{\bar{x}}). \end{aligned}$$

(83)

Having the classical Poincaré inequality and (81), we obtain that

(84)

Let us set

By (76) we have the weak convergence of $|{{D{\bar{\varvec{\textsf {u}}}}}_j}-{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|\rightarrow {\mathfrak {h}}$ in $L^{H_s}(B_\varrho ).$ Since ${\bar{x}}$ is a Lebesgue’s point of ${D{{\widetilde{\varvec{\textsf {u}}}}}}$ we infer that

$$\begin{aligned} {\mathfrak {h}}({\bar{x}})=\lim _{\varrho \rightarrow 0}\lim _{j\rightarrow \infty }\text {I}_{j,\varrho }^0. \end{aligned}$$

(85)

In order to prove that ${\mathfrak {h}}({\bar{x}})=0$, let us write

(86)

and prove the convergence of both terms first when $j\rightarrow \infty$ and then $\varrho \rightarrow 0$.

We start with $\text {I}^1_{j,\varrho }$. Let us observe that

Notice that $\text {I}^{1,1}_{j,\varrho }$ vanishes as $j\rightarrow \infty$. Indeed, $\text {I}^{1,1}_{j,\varrho }\ge 0$ and by Hölder inequality we have

$$\begin{aligned} \text {I}^{1,1}_{j,\varrho }&\le \frac{1}{|B_{\varrho /2}({\bar{x}})|} \left( \int _{B_{\varrho /2}({\bar{x}})}|{{D{\bar{\varvec{\textsf {u}}}}}_j}-{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|^{1+\epsilon }\,dx\right) ^{\frac{1}{1+\epsilon }}\left( |\{x\in B_{3/4}:\ |{\bar{\varvec{\textsf {u}}}}_j-{{{\widetilde{\varvec{\textsf {u}}}}}}|\ge \varrho /2\}|\right) ^\frac{\epsilon }{1+\epsilon }. \end{aligned}$$

Since by (76) one has that $|{\bar{\varvec{\textsf {u}}}}_j-{{{\widetilde{\varvec{\textsf {u}}}}}}|\rightarrow 0$ strongly in $L^1(B_{3/4})$, so

$$\begin{aligned} \lim _{j\rightarrow \infty } |\{x\in B_{3/4}:\ |{\bar{\varvec{\textsf {u}}}}_j-{{{\widetilde{\varvec{\textsf {u}}}}}}|\ge \varrho /2\}|=0. \end{aligned}$$

The rest of the terms are bounded as $\epsilon$ is chosen such that (75) is true and $|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|$ shares the same a priori estimates as $|{D{\bar{\varvec{\textsf {u}}}}}_j|$. Therefore, we infer that $\lim _{j\rightarrow \infty } \text {I}^{1,1}_{j,\varrho }=0$. On the other hand, $\text {I}^{1,2}_{j,\varrho }$ is convergent when $j\rightarrow \infty$, because of the weak convergence of $|{{D{\bar{\varvec{\textsf {u}}}}}_j}-{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|\rightarrow {\mathfrak {h}}$ in $L^{H_s}(B_\varrho )$. Hence, we get

We can estimate further

which tends to 0 as $\varrho \rightarrow 0$ as ${\bar{x}}$ is a Lebesgue’s point of ${\mathfrak {h}}$ as in (81) and the last bracket converges to 0 due to (84). Altogether, we have that $\text {I}^{1}_{j,\varrho }$ vanishes in the limit, so we will now concentrate on $\text {I}^2_{j,\varrho }$ for which we have

By (76), (81), and (83) we have that $\lim _{\varrho \rightarrow 0}\limsup _{j\rightarrow \infty }\text {I}^{2,2}_{j,\varrho }=0.$ Proving the convergence

$$\begin{aligned} \limsup _{\varrho \rightarrow 0}\limsup _{j\rightarrow \infty }\text {I}^{2,1}_{j,\varrho }=0 \end{aligned}$$

(87)

requires more arguments. We take

$$\begin{aligned} \phi \in C_c^\infty (B_\varrho ({{\bar{x}}}))\quad \,\text {with}\, 0\le \phi \le 1, \ \phi \equiv 1 \,\text {on}\, B_{\varrho /2}({{\bar{x}}})\ \,\text {and}\, \ |D\phi |\le 4/\varrho . \end{aligned}$$

Let

$$\begin{aligned} {{\varvec{{\mathsf{\eta }}}}}=\phi T_\varrho ({{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}), \end{aligned}$$

where the truncation is defined in (17). Let us denote

$$\begin{aligned} {\textsf{P}_\textsf{j}}:=\frac{({{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}})\otimes ({{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}})}{|{{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|^2}\qquad \text {and}\qquad {{\textsf{P}}}:=\frac{({{{\widetilde{\varvec{\textsf {u}}}}}}-{{\varvec{{\mathsf{\ell _\varrho }}}}})\otimes ({{{\widetilde{\varvec{\textsf {u}}}}}}-{{\varvec{{\mathsf{\ell _\varrho }}}}})}{|{{{\widetilde{\varvec{\textsf {u}}}}}}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|^2} \end{aligned}$$

(88)

when $|{{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|\ne 0$ and $|{{{\widetilde{\varvec{\textsf {u}}}}}}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|\ne 0,$ respectively. Within this notation we have that

$$\begin{aligned}&\left( {\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j})-{\bar{{{\mathcal {A}}}}}(x,D{{{\varvec{{\mathsf{\ell _\varrho }}}}}})\right) :{D{{\varvec{{\mathsf{\eta }}}}}}\\&\quad =\mathbb {1}_{\{|{{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|<\varrho \}}\left[ \left( {\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j})-{\bar{{{\mathcal {A}}}}}(x,D{{{\varvec{{\mathsf{\ell _\varrho }}}}}})\right) :{D({{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}})}\right] \phi \\&\qquad +\frac{\varrho \mathbb {1}_{\{|{{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|\ge \varrho \}}}{|{{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|}\left[ \left( {\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j})-{\bar{{{\mathcal {A}}}}}(x,D{{{\varvec{{\mathsf{\ell _\varrho }}}}}})\right) :({\textsf{Id}}-{\textsf{P}_\textsf{j}}){D({{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}})}\right] \phi \\&\qquad +\left( {\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j})-{\bar{{{\mathcal {A}}}}}(x,D{{{\varvec{{\mathsf{\ell _\varrho }}}}}})\right) :\left[ T_\varrho ({{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}})\otimes D\phi \right] \\&\quad =: G_{j,\varrho }^{1}(x)+G_{j,\varrho }^2(x)+G_{j,\varrho }^3(x) . \end{aligned}$$

Moreover, we recall that since ${{\varvec{{\mathsf{\ell _\varrho }}}}}$ is affine, whenever $B\subset B_1$ it holds that

$$\begin{aligned} \int _B{\bar{{{\mathcal {A}}}}}(x,D{{{\varvec{{\mathsf{\ell _\varrho }}}}}}):D{{{\varvec{{\mathsf{\varphi }}}}}}\,dx=0\qquad \text {for every }\ {{\varvec{{\mathsf{\varphi }}}}}\in W^{1,1}_0(B,{{\mathbb {R}}^{m}}). \end{aligned}$$

Therefore, by (72) it is justified to write that

(89)

The first term in the above display is nonnegative because of the monotonicity of ${\bar{{{\mathcal {A}}}}}.$ Instrumental for proving that $\text {I}_{j,\varrho }^{2,1}\rightarrow 0$ is to establish that

(90)

which will be proven provided one justifies that the last two terms of (89) vanish in the limit. We will show first that

(91)

The quasi-diagonal structure of ${\bar{{{\mathcal {A}}}}}$ ensures that ${\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j}):[({\textsf{Id}}-{\textsf{P}_\textsf{j}}){{D{\bar{\varvec{\textsf {u}}}}}_j}]\ge 0$, see (49). Therefore,

$$\begin{aligned}&\big ({\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j})-{\bar{{{\mathcal {A}}}}}(x,D{{{\varvec{{\mathsf{\ell _\varrho }}}}}})\big ):({\textsf{Id}}-{\textsf{P}_\textsf{j}}){D({{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}})}\nonumber \\&\quad \ge -{\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j}):({\textsf{Id}}-{\textsf{P}_\textsf{j}})D{{{\varvec{{\mathsf{\ell _\varrho }}}}}}-{\bar{{{\mathcal {A}}}}}(x,D{{{\varvec{{\mathsf{\ell _\varrho }}}}}}):({\textsf{Id}}-{\textsf{P}_\textsf{j}}){D({{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}})}. \end{aligned}$$

(92)

Recall that ${\textsf{P}_\textsf{j}}$ and ${{\textsf{P}}}$, defined in (88), are bounded. Notice that for $j\rightarrow \infty$ we have $\mathbb {1}_{\{|{{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|\ge \varrho \}}{\textsf{P}_\textsf{j}}\rightarrow \mathbb {1}_{\{|{{{\widetilde{\varvec{\textsf {u}}}}}}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|\ge \varrho \}}{{\textsf{P}}}$ almost everywhere and thus, by the Lebesgue’s dominated convergence theorem, also strongly in $L^t(B_{3/4})$ for every $t\ge 1.$ Moreover, $\mathbb {1}_{\{|{{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|\ge \varrho \}}|{{\bar{\varvec{\textsf {u}}}}_j}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|^{-1}\rightarrow \mathbb {1}_{\{|{{{\widetilde{\varvec{\textsf {u}}}}}}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|\ge \varrho \}}|{{{\widetilde{\varvec{\textsf {u}}}}}}-{{\varvec{{\mathsf{\ell _\varrho }}}}}|^{-1}$ almost everywhere and, as a uniformly bounded sequence of functions, it converges also strongly in $L^t(B_{3/4})$ for every $t\ge 1.$ Having this, (92), and (76), we obtain

(93)

We can estimate

where the first term is bounded and the second term is convergent to zero by (84). On the other hand, by (83) and (82) we may estimate

where, again, the first term is bounded and the second convergent to zero by (84). Summing up the information from the last three displays we get (91).

Now we concentrate on justifying that

(94)

Let us observe that because (76) provides weak convergence of ${\bar{{{\mathcal {A}}}}}(x,{{D{\bar{\varvec{\textsf {u}}}}}_j})$ to ${{{\mathfrak {A}}}}$ in $L^{1+\epsilon }$ and strong convergence ${{\bar{\varvec{\textsf {u}}}}_j}$ to ${{{\widetilde{\varvec{\textsf {u}}}}}}$ in $L^{1+\epsilon }$, we obtain that

By Hölder inequality and the choice of $\phi$, we estimate further

where the first integral on the right-hand side is finite and the second term converges to zero. Indeed, since $0 \le \min \{ \varrho , |{{{\widetilde{\varvec{\textsf {u}}}}}} - {{\varvec{{\mathsf{\ell _\varrho }}}}}|\} \le \varrho$, we have

where the last convergence results from (84). Therefore, we get (94).

We have shown (91) and (94), so because of (89) the limit (90) follows. Hence, we are in the position to prove (87). In the view of (21), (90) implies that

(95)

At this stage, we calculate similarly to (80) in order to show (87). For any ${\hat{\epsilon }}<\frac{2}{s_{G}}$, it is readily checked that $t \mapsto t g(t)^{-{\hat{\epsilon }}/(2- {\hat{\epsilon }})}$ is a monotone increasing function. Then

where $c=c(g)>0$. Noting that the very last term in the above display is bounded, by (95) we infer that (87) holds.

Summing up all the convergences of this step, we get in (85) that ${\mathfrak {h}}({\bar{x}})=0$ and, consequently, (79) holds almost everywhere in $B_{3/4}.$ As explained in the beginning of this step, this suffices to get the final aim of Step 5, that is strong convergence of gradients (78).

Step 6. ${\bar{{{\mathcal {A}}}}}$-harmonicity of the limit map and conclusion by contradiction. Having (78), we can pass to the limit in (72) with $j\rightarrow \infty$ getting that

(96)

Therefore, if ${{{\widetilde{\varvec{\textsf {u}}}}}}\in W^{1,G}(B_{1/2},{{\mathbb {R}}^{m}})$, then it will be proven to be ${\bar{{{\mathcal {A}}}}}$-harmonic. Indeed, since we know (77), it is allowed to take ${\bar{\varvec{\textsf {{v}}}}}={{{\widetilde{\varvec{\textsf {u}}}}}}$ in (73). Note that in such a case (77) is precisely the restriction on the test function from (74). Then, in the view of (78), taking j large enough, we will get the desired contradiction. Hence, it remains to prove that $|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|\in L^G(B_{1/2})$.

We have (63) for each ${{\bar{\varvec{\textsf {u}}}}_j}$ with the constant independent of j, so by the lower semicontinuity we can write that

$$\begin{aligned} \left( \int _{B_{7/8}}\left( \frac{{{\bar{G}}}(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)}{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{1+{\widetilde{\gamma }}}}\right) ^\frac{n}{n-\kappa }\,dx\right) ^\frac{n-\kappa }{n}&\le \liminf _{j\rightarrow \infty }\left( \int _{B_{7/8}}\left( \frac{{{\bar{G}}}(1+|{{\bar{\varvec{\textsf {u}}}}_j}|)}{(1+|{{\bar{\varvec{\textsf {u}}}}_j}|)^{1+{\widetilde{\gamma }}}}\right) ^\frac{n}{n-\kappa }\,dx\right) ^\frac{n-\kappa }{n}\\&\le c \end{aligned}$$

for some $c=c({{\texttt {\textit{data}}}},{\widetilde{\gamma }})>0.$ Analogically, by (50) for ${{\bar{\varvec{\textsf {u}}}}_j}$ and with $\delta =2^{-j}$, by letting $j\rightarrow \infty$, Fatou’s lemma on the left-hand side of the resultant inequality and (78) on its right-hand side, we get

$$\begin{aligned} \int _{B_{3/4}\cap \{|{{{\widetilde{\varvec{\textsf {u}}}}}}|< t\}}{{\bar{G}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|) \phi ^q\,dx&\le \,c_* \int _{B_{3/4}\cap \{|{{{\widetilde{\varvec{\textsf {u}}}}}}|< t\}}{{\bar{G}}}(|{{{\widetilde{\varvec{\textsf {u}}}}}}|\,|D\phi |) \,dx\\&\quad +\,c_*\,t \int _{B_{3/4}\cap \{|{{{\widetilde{\varvec{\textsf {u}}}}}}|\ge t\}} {{{\bar{g}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)}\phi ^{q-1}| D\phi |\,dx \end{aligned}$$

for every $t>0$ and $\phi \in C_c^\infty (B_{3/4})$ with $\phi \ge 0$, and $c_*=c_*({{\texttt {\textit{data}}}},q).$ We proceed as in the beginning of Step 3. We multiply the above display by $(1+t)^{-({\widetilde{\gamma }}+1)},$ ${\widetilde{\gamma }}>0$ to be chosen sufficiently small in a few lines, integrate it from zero to infinity and apply Cavalieri’s principle (Lemma 3.6) twice (with $\nu _1={{\bar{G}}}(|{D{{\widetilde{\varvec{\textsf {u}}}}}}|) \phi ^q$ and $\nu _2={{\bar{G}}}(|{{\widetilde{\varvec{\textsf {u}}}}}|\,|D\phi |)$. Altogether we get

$$\begin{aligned}&\frac{1}{{\widetilde{\gamma }}} \int _{B_{3/4}}\frac{{{\bar{G}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|) \phi ^q}{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\,dx\le \frac{c_*}{{\widetilde{\gamma }}} \int _{B_{3/4}}\frac{{{\bar{G}}}(|{{{\widetilde{\varvec{\textsf {u}}}}}}|\,|D\phi |) }{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\,dx\\&\quad + c_* \int _0^\infty \frac{1}{(1+t)^{{\widetilde{\gamma }}}}\int _{B_{3/4}\cap \{|{{{\widetilde{\varvec{\textsf {u}}}}}}|\ge t\} } {{{\bar{g}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)}\phi ^{q-1}| D\phi |\,dx \,dt=\text {III}_1+\text {III}_2. \end{aligned}$$

To estimate further the very last term we note that q is large enough to satisfy $s_G'\ge q'$, and so Lemma 3.1 implies that ${{{\bar{G}}}}^*(\phi ^{q-1}{{\bar{g}}}(t))\le c_G \phi ^q {{\bar{G}}}(t).$ Then, using Young inequality (20) and by taking ${\widetilde{\gamma }}\in (0,1/(2c_*c_G+1)],$ get

$$\begin{aligned} \begin{aligned} \text {III}_2&\le \frac{c_*}{1-{\widetilde{\gamma }}}\int _{B_{3/4} } {{{\bar{g}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)}{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{1-{\widetilde{\gamma }}}}\phi ^{q-1}| D\phi |\,dx\\&\le \frac{1}{2{\widetilde{\gamma }}c_G} \int _{B_{3/4}}\frac{{{{\bar{G}}}^*}\left( {\phi ^{q-1}{{\bar{g}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|) }\right) }{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}} \, dx+ \frac{1}{2{\widetilde{\gamma }}}\int _{B_{3/4}}\frac{{{\bar{G}}}\left( \frac{2{\widetilde{\gamma }}c_* c_G}{1-{\widetilde{\gamma }}}(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)|D\phi |\right) }{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\,dx \\&\le \frac{1}{2{\widetilde{\gamma }}} \int _{B_{3/4}}\frac{{{\bar{G}}}\left( |{{D{{\widetilde{\varvec{\textsf {u}}}}}}}| \right) \phi ^{q }}{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}} \,dx + \frac{c_* c_G}{1- {\widetilde{\gamma }}} \int _{B_{3/4}}\frac{{{\bar{G}}}\left( (1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|) |D\phi |\right) }{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\, dx \end{aligned} \end{aligned}$$

where in the last line we used that $\frac{2{\widetilde{\gamma }}c_* c_G}{1-{\widetilde{\gamma }}}<1$ can be taken out of the integrand by Jensen’s inequality. Summing up we obtain

$$\begin{aligned} \int _{B_{3/4} }\frac{{{\bar{G}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|) \phi ^q}{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\,dx&\le \,{C} \int _{B_{3/4}}\frac{{{\bar{G}}}\left( (1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)|D\phi |\right) }{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\, dx \end{aligned}$$

(97)

where $C=C({{\texttt {\textit{data}}}}) >0$. Then similar calculations to (55)-(59) yield that

$$\begin{aligned} \left( \int _{B_{3/4}}\left( \frac{{{\bar{G}}}((1+|{\bar{\varvec{\textsf {u}}}}|)\phi ^q)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{{\widetilde{\gamma }}}}\right) ^\frac{n}{n-1}\,dx\right) ^\frac{n-1}{n}\le c \int _{B_{3/4}}\frac{{{\bar{G}}}\left( (1+|{\bar{\varvec{\textsf {u}}}}|)|D\phi |\right) }{(1+|{\bar{\varvec{\textsf {u}}}}|)^{{\widetilde{\gamma }}}}\, dx. \end{aligned}$$

holds with $c=c({{\texttt {\textit{data}}}})>0$. Indeed, in Step 4, we have checked that the above $c=c({{\texttt {\textit{data}}}},{\widetilde{\gamma }})$ is an increasing function of ${\widetilde{\gamma }}$. As we consider small ${\widetilde{\gamma }}$, c in fact depends only on ${{\texttt {\textit{data}}}}$. For $5/8 \le r_{1} < r_{2} \le 3/4$ we take $\phi \in C_{c}^{\infty }(B_{r_2})$ to satisfy

$$\begin{aligned} \phi \equiv 1 \quad \text {on} \quad B_{r_1} \qquad \text {and} \qquad |{D \phi }| \le \frac{100}{r_2 - r_1}. \end{aligned}$$

Then the doubling property of ${{\bar{G}}}$ and Lemma 3.9 gives

$$\begin{aligned} \left( \int _{B_{5/8}}\left( \frac{{{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{{\widetilde{\gamma }}}}\right) ^\frac{n}{n-1}\,dx\right) ^\frac{n-1}{n}\le c \left( \int _{B_{3/4}} \left( \frac{{{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{{\widetilde{\gamma }}}} \right) ^{\frac{1}{2s_G}}\, dx \right) ^{2s_G}. \end{aligned}$$

(98)

We now restrict ourselves to ${\widetilde{\gamma }}\in \big (0,\min \{1/(2c_*c_G+1), \frac{i_G}{2}\}\big )$ and define

$$\begin{aligned} \Psi _{{\widetilde{\gamma }}}(t) = \int _{0}^{t} \frac{1}{\tau } \left( \frac{{{\bar{G}}}(\tau )}{\tau ^{{\widetilde{\gamma }}}} \right) ^{\frac{1}{2s_G}} \, d \tau , \end{aligned}$$

which is an increasing concave function on $[0,\infty )$ satisfying

$$\begin{aligned} \frac{t \Psi _{{\widetilde{\gamma }}}''(t)}{\Psi _{{\widetilde{\gamma }}}'(t)}=\frac{t g(t)}{2s_G {{\bar{G}}}(t)} - 1 - \frac{{\widetilde{\gamma }}}{2s_G} \in (-1,-1/2) \quad \text {and} \quad \Psi _{{\widetilde{\gamma }}}(t) \approx \left( \frac{{{\bar{G}}}(t)}{t^{{\widetilde{\gamma }}}}\right) ^{\frac{1}{2s_G}}. \end{aligned}$$

Then Jensen’s inequality gives

$$\begin{aligned}&\int _{B_{3/4}} \left( \frac{{{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{{\widetilde{\gamma }}}} \right) ^{\frac{1}{2s_G}}\, dx \le c \int _{B_{3/4}} \Psi _{{\widetilde{\gamma }}}(1+|{\bar{\varvec{\textsf {u}}}}|)\, dx \le c\, \Psi _{{\widetilde{\gamma }}} \left( \int _{B_{3/4}} (1+|{\bar{\varvec{\textsf {u}}}}|) \, dx \right) \nonumber \\&\qquad \quad \le c\, \Psi _{{\widetilde{\gamma }}} \left( \int _{B_{3/4}} (1+|{\bar{\varvec{\textsf {u}}}}|) \, dx + 1\right) \le c\, {{\bar{G}}}^\frac{1}{2s_G} \left( \int _{B_{3/4}} (1+|{\bar{\varvec{\textsf {u}}}}|) \, dx + 1\right) \le c, \end{aligned}$$

(99)

with $c=c({{\texttt {\textit{data}}}})>0.$ We used that if $t>1$ is arbitrarily fixed there exists $c>0$ independent of ${\widetilde{\gamma }}$ such that for all $t>1$ and all ${\widetilde{\gamma }}$, we have $\Psi _{\widetilde{\gamma }}(t)\le c \left( {{{\bar{G}}}(t)} \right) ^{\frac{1}{2s_G}}$. By Hölder inequality, (98) and (99) we get that

$$\begin{aligned} \int _{B_{5/8}}\frac{{{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{{\widetilde{\gamma }}}}\,dx\le \left( \int _{B_{5/8}}\left( \frac{{{\bar{G}}}(1+|{\bar{\varvec{\textsf {u}}}}|)}{(1+|{\bar{\varvec{\textsf {u}}}}|)^{{\widetilde{\gamma }}}}\right) ^\frac{n}{n-1}\,dx\right) ^\frac{n-1}{n}\le c \end{aligned}$$

(100)

with $c=c({{\texttt {\textit{data}}}})>0.$

We now consider (97) with a cutoff function $\phi \in C_{c}^{\infty }(B_{5/8})$ satisfying

$$\begin{aligned} \phi \equiv 1 \quad \text {in} \quad B_{1/2} \qquad \text {and} \qquad |{D \phi }| \le 100 \end{aligned}$$

and combine it with (100), to obtain for some $c=c({{\texttt {\textit{data}}}})>0$ that

$$\begin{aligned} \int _{B_{1/2} }\frac{{{\bar{G}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)}{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\,dx&\le {c} \int _{B_{3/4}}\frac{{{\bar{G}}}(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)}{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\, dx \le c. \end{aligned}$$

Therefore, using Fatou’s lemma we justify that

$$\begin{aligned} \int _{B_{1/2} }{{{\bar{G}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)}\,dx&\le \limsup _{{\widetilde{\gamma }}\rightarrow 0} \int _{B_{1/2} }\frac{{{\bar{G}}}(|{{D{{\widetilde{\varvec{\textsf {u}}}}}}}|)}{(1+|{{{\widetilde{\varvec{\textsf {u}}}}}}|)^{{\widetilde{\gamma }}}}\,dx\le c. \end{aligned}$$

Consequently, we conclude that $|{D {{{\widetilde{\varvec{\textsf {u}}}}}}}| \in L^{G}(B_{1/2})$. This completes the proof of Theorem 4.1. $\square$

5 Proof of Wolff potential estimates

5.1 Comparison estimate

We need one more auxiliary estimate yielding comparison between energy of a weak solution and an ${{\mathcal {A}}}$-harmonic function.

Lemma 5.1

Under Assumption (A-vect) suppose $u\in W^{1,G}(B_r,{{\mathbb {R}}^{m}})$ is a weak solution to (4) in $B_r=B_r(x_0),$ $r<1$ and let $\varepsilon \in (0,1)$. Then there exists a positive constant $c_{\textrm{s}}=c_{\textrm{s}}({{\texttt {\textit{data}}}},\varepsilon )$ and a map ${\varvec{\textsf {v}}}$ being ${{\mathcal {A}}}$-harmonic in $B_{r/2}$ and such that

(101)

Proof

Let us fix

with $\delta =\delta ({{\texttt {\textit{data}}}},\varepsilon )$ from Theorem 4.1 with $M=1$. If $\lambda =0,$ then ${\varvec{\textsf {u}}}$ is constant and ${\varvec{\textsf {v}}}={\varvec{\textsf {u}}}$. Otherwise $\lambda >0$ and we can argue by scaling

$$\begin{aligned} {\bar{\varvec{\textsf {u}}}}:=\frac{{\varvec{\textsf {u}}}-({\varvec{\textsf {u}}})_{B_r}}{\lambda },\qquad \bar{{{\varvec{{\mathsf{\mu }}}}}}:=\frac{{{\varvec{{\mathsf{\mu }}}}}}{g(\lambda )},\qquad {\bar{{{\mathcal {A}}}}}(x,{{\varvec{{\mathsf{\xi }}}}})=\frac{{{\mathcal {A}}}(x,\lambda {{\varvec{{\mathsf{\xi }}}}})}{g(\lambda )}. \end{aligned}$$

Then

By definition of ${\bar{\varvec{\textsf {u}}}}$ and $\lambda$ we notice that

Therefore, by Theorem 4.1 applied to ${\bar{\varvec{\textsf {u}}}}$ we get that there exists ${\bar{\varvec{\textsf {{v}}}}}$ being ${\bar{{{\mathcal {A}}}}}$-harmonic in $B_{r/2}$ and such that

Then (101) follows by rescaling back with ${\varvec{\textsf {v}}}=\lambda {\bar{\varvec{\textsf {{v}}}}}$ which is ${{\mathcal {A}}}$-harmonic. $\square$

5.2 Estimates on concentric balls

This subsection is devoted to prove some properties of weak solutions to (4) with ${{\varvec{{\mathsf{\mu }}}}}\in C^\infty (\Omega ,{{\mathbb {R}}^{m}})$ holding over a family of concentric balls $\{B^j\}$. Before we pass to this, let us fix some notation and parameters. Recall that we have chosen $R_0 = R_0({{\texttt {\textit{data}}}}, \varsigma )$ in Proposition 3.13. We take an arbitrary constant $\alpha _V \in (0,1)$ and take $\varsigma =\varsigma (s_G,\alpha _V)$ to satisfy $\alpha _{D}:=\frac{\alpha _V+1}{2} \le 1+(\varsigma -1){K}$. To prove Theorem 2.1, it is enough to take $\alpha _V=\frac{1}{2}$, but for the later use in the proof of Theorem 2.9, we have taken $\alpha _V$ arbitrarily. We now choose

$$\begin{aligned} \sigma _0:= \min \left\{ \left( \frac{1}{2^{n+6} c_{\textrm{o}}}\right) ^{\frac{1-\alpha _V}{2}},\frac{1}{4}\right\} . \end{aligned}$$

(102)

If $r\in (0,R_0)$ is given, for every $j\in {{\mathbb {N}}}\cup \{0\}$ let us fix

$$\begin{aligned} r_j:=\sigma ^{j+1}r,\qquad \qquad B^j:=\overline{B_{r_j}(x_0)}, \end{aligned}$$

so that $r_{-1}=r.$ We denote

(103)

Lemma 5.2

Suppose Assumption (A-vect) is satisfied. If ${\varvec{\textsf {u}}}\in W^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ is a weak solution to (4) with ${{\varvec{{\mathsf{\mu }}}}}\in C^\infty (\Omega ,{{\mathbb {R}}^{m}})$, $j\in {{\mathbb {N}}}$ is fixed, $E_j$ is given by (103), while $0<\sigma \le \sigma _0$ is arbitrary, then we have that

$$\begin{aligned} E_{j+1}\le c_{\textrm{D}}\sigma ^{\alpha _D}E_j+c_{\textrm{E}}r_jg^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B^j)}{r_j^{n-1}}\right) \end{aligned}$$

(104)

for $c_{\textrm{D}}=c_{\textrm{D}}({{\texttt {\textit{data}}}},\alpha _V) =2^{n+4}c_{\textrm{o}}$ and $c_{\textrm{E}}=c_{\textrm{E}}({{\texttt {\textit{data}}}},\alpha _V)={2^{n+2}c_{\textrm{s}}c_{\textrm{P}}c_{\textrm{o}}}\sigma ^{-n},$ where $c_{\textrm{P}}$ is the constant from Poincaré inequality in $W^{1,1}(\Omega ,{{\mathbb {R}}^{m}})$.

Proof

We may apply Lemma 5.1 in $B_r=B_{r_j}(x_0)$ to get that there exists an ${{\mathcal {A}}}$-harmonic map ${\varvec{\textsf {v}}}_j\in W^{1,G}(B_{r_{j/2}},{{\mathbb {R}}^{m}})$ in $B_{r_{j/2}}$ such that

(105)

By Poincaré inequality in $W^{1,1}(\Omega ,{{\mathbb {R}}^{m}})$ for ${\varvec{\textsf {w}}}_j={\varvec{\textsf {u}}}-{\varvec{\textsf {v}}}_j$ we have

Then

(106)

Thus by Lemma 3.5, the triangle inequality, (106), and Proposition 3.13, we estimate

By choosing $\varepsilon =\frac{\sigma ^{n+\alpha _V}}{c_{\textrm{P}}}$ we complete the proof. $\square$

Proposition 5.3

Suppose Assumption (A-vect) is satisfied. If ${\varvec{\textsf {u}}}\in W^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ is a weak solution to (4) with ${{\varvec{{\mathsf{\mu }}}}}\in C^\infty (\Omega ,{{\mathbb {R}}^{m}})$, then there exists a constant $c_V=c_V({{\texttt {\textit{data}}}},\alpha _V)\ge 1$ such that for every $\tau \in (0,1]$ we have

Proof

Lemma 5.2 implies that for $j\in {{\mathbb {N}}}\cup \{0\}$ it holds that

$$\begin{aligned} E_{j+1}\le c_{\textrm{D}}\sigma ^{\alpha _D}E_j+c_{\textrm{E}}r_jg^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B^j)}{r_j^{n-1}}\right) . \end{aligned}$$

Iterating this estimate we get that for any $k \in {{\mathbb {N}}}\cup \{0\}$ we have

$$\begin{aligned} E_{k+1}\le (c_{D}\sigma ^{\alpha _D})^{k+1} E_0+c_{\textrm{E}} \sum _{j=0}^k (c_{\textrm{D}} \sigma ^{\alpha _{\textrm{D}}})^{j}r_j\,g^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B^j)}{r_j^{n-1}}\right) , \end{aligned}$$

where $c=c({{\texttt {\textit{data}}}})$. Recalling $\alpha _{D}=\frac{\alpha _V+1}{2}$, $c_{\textrm{D}}=2^{n+4}c_{\textrm{o}}$ and (102), we see

$$\begin{aligned} c_{\textrm{D}} \sigma ^{\alpha _{\textrm{D}}} \le \frac{\sigma ^{\alpha _V}}{4}. \end{aligned}$$

By Lemma 3.5 and direct computation we have for any $k\in {{\mathbb {N}}}\cup \{0\}$ that

(107)

We take $\tau \in (0,\sigma )$ and $k\ge 1$ such that $\sigma ^{k+1}<\tau \le \sigma ^k.$ Then by Lemma 3.5 and (107) we obtain

By taking $c_V=c_V({{\texttt {\textit{data}}}},\alpha _V)=2c_{\textrm{E}}\sigma ^{-n-2\alpha _V}$ we conclude the claim for $\tau \in (0,\sigma ).$ For completing the range of $\tau \in [\sigma ,1]$ it suffices to note that

$\square$

5.3 SOLA ${\varvec{\textsf {u}}}$ belongs to VMO

Proof of Proposition 2.2

Suppose ${{\varvec{{\mathsf{\mu }}}}}\in C^\infty (\Omega ,{{\mathbb {R}}^{m}})$ and ${\varvec{\textsf {u}}}\in W^{1,G}(\Omega ,{{\mathbb {R}}^{m}})$ is a weak solution to (4). By Proposition 5.3 we can find constants $c_V=c_V({{\texttt {\textit{data}}}},\alpha _V)$ we have

(108)

Let us consider a SOLA ${\varvec{\textsf {u}}}\in W^{1,1}(\Omega ,{{\mathbb {R}}^{m}})$ existing due to Proposition 3.4. Suppose $({\varvec{\textsf {u}}}_h)$ and $({{\varvec{{\mathsf{\mu }}}}}_h)$ are approximating sequences from definition of SOLA, see Sect. 2.2. Inequality (108) hold for each ${\varvec{\textsf {u}}}_h$ and ${{\varvec{{\mathsf{\mu }}}}}_h$. We have to motivate passing to the limit with $h\rightarrow \infty$. Since (7) holds, we can write (108) for the original SOLA too. From now on this kind of solution is considered.

Our aim now is to show that ${\varvec{\textsf {u}}}$ is VMO at $x_0$ provided (10) is assumed. Let $\delta \in (0,1)$. By (10) we find a positive radius $r_{1,\delta }<r$ such that

$$\begin{aligned} c_V\sup _{0<\varrho <r_{1,\delta }}\varrho \, g^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B_\varrho (x_0))}{\varrho ^{n-1}}\right) \le \frac{\delta }{2} \end{aligned}$$

and then $\tau _\delta$ so small that

For $r_\delta :=\tau _\delta r_{1,\delta }$ from estimate (108) (applied with $r=r_{1,\delta }$) it follows that

that is that ${\varvec{\textsf {u}}}$ has vanishing mean oscillation at $x_0$. $\square$

5.4 Proofs of Theorems 2.1, 2.3 and 2.9

We start with the proof of pointwise Wolff potential estimate, then pass to continuity and Hölder continuity criteria.

Proof of Theorem 2.1

We notice that having $E_j$ defined in (103) with $r=r^j$ we can fix $\sigma$ in Lemma 5.2 to get that

$$\begin{aligned} E_{j+1}\le \frac{1}{2}E_j+cr^jg^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B^j)}{r_j^{n-1}}\right) \qquad \text {for every }\ j\in {{\mathbb {N}}}\cup \{0\}. \end{aligned}$$

(109)

We sum up inequalities from (109) to obtain

$$\begin{aligned} \sum _{j=1}^{k+1} E_j\le \frac{1}{2}\sum _{j=0}^k E_j+c\sum _{j=0}^k r_jg^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B^j)}{r_j^{n-1}}\right) ,\qquad k\in {{\mathbb {N}}}\cup \{0\}. \end{aligned}$$

By rearranging terms we have

$$\begin{aligned} \sum _{j=1}^{k+1} E_j\le 2 E_0+c\sum _{j=0}^k r_jg^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B^j)}{r_j^{n-1}}\right) . \end{aligned}$$

We notice that for some $c=c({{\texttt {\textit{data}}}})$ we can estimate

$$\begin{aligned} \sum _{j=0}^k r_jg^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B^j)}{r_j^{n-1}}\right) \le c\,\int _0^r g^{-1}\left( \frac{|{{\varvec{{\mathsf{\mu }}}}}|(B_\varrho )}{\varrho ^{n-1}}\right) \,d\varrho =c\,{{\mathcal {W}}}^{{\varvec{{\mathsf{\mu }}}}}_G(x_0,r). \end{aligned}$$

Last two displays imply that

$$\begin{aligned} \sum _{j=1}^{k+1} E_j\le 2 E_0+c{{\mathcal {W}}}^{{\varvec{{\mathsf{\mu }}}}}_G(x_0,r). \end{aligned}$$

For every $m,k\in {{\mathbb {N}}}$ such that $m<k$ we have

where $\sigma =\sigma ({{\texttt {\textit{data}}}})$ and $c=c({{\texttt {\textit{data}}}}).$ For $j\rightarrow \infty$, $(({\varvec{\textsf {u}}})_{B^j})_j$ is a Cauchy sequence that converges to ${\varvec{\textsf {u}}}(x_0)$, that is

$$\begin{aligned} \lim _{\varrho \rightarrow 0}({\varvec{\textsf {u}}})_{B_\varrho (x_0)}={\varvec{\textsf {u}}}(x_0) \end{aligned}$$

and $x_0$ is a Lebesgue’s point of ${\varvec{\textsf {u}}}$. This completes the proof of (8), while (9) follows as a direct corollary.$\square$

Let us concentrate on the continuity criterion.

Proof of Theorem 2.3

Our aim is to infer continuity of ${\varvec{\textsf {u}}}$ in $B_r(x_0)$ knowing that (12) holds. We will show that for every $\delta >0$ and $x_1\in B_r(x_0)$ we can find $r_\delta \in (0,\textrm{dist}\,(B_r(x_0),\partial \Omega ))$ such that

$$\begin{aligned} \textrm{osc}_{B_{r_\delta }(x_1)}\,{\varvec{\textsf {u}}}<\delta . \end{aligned}$$

(110)

Without loss of generality we assume that ${{\varvec{{\mathsf{\mu }}}}}$ is defined on whole ${{\mathbb {R}}^{n}}$, as we can extend it by zero outside $\Omega$. By (12) we can take $\varrho _1$ small enough for

$$\begin{aligned} \sup _{x\in B_r(x_0)}{{\mathcal {W}}}^{{\varvec{{\mathsf{\mu }}}}}_G(x,\varrho _1)\le \frac{\delta }{16}. \end{aligned}$$

(111)

Let $r_\delta >0$ to be chosen in a moment. We take an arbitrary point $x_2\in B_{r_\delta }(x_1)$ and estimate

$$\begin{aligned} |{\varvec{\textsf {u}}}(x_1)-{\varvec{\textsf {u}}}(x_2)|&\le |{\varvec{\textsf {u}}}(x_1)-({\varvec{\textsf {u}}})_{B_{2r_\delta }(x_1)}|+|({\varvec{\textsf {u}}})_{B_{2r_\delta }(x_1)}-({\varvec{\textsf {u}}})_{B_{r_\delta }(x_2)}|\nonumber \\&\quad +|({\varvec{\textsf {u}}})_{B_{r_\delta }(x_2)}-{\varvec{\textsf {u}}}(x_2)|=:A_1+A_2+A_3. \end{aligned}$$

(112)

We start with estimating $A_2$ by noting that

Since (12) implies (10), Proposition 2.2 implies that ${\varvec{\textsf {u}}}$ has vanishing mean oscillations at $x_1$. Therefore there exists $\varrho _2\in (0,\min \{\varrho _1,\textrm{dist}(x_1,\partial B_r(x_0))/4\})$ such that for every $\varrho \le \varrho _2$ it holds

We choose $r_\delta =\varrho _2/2$ and observe that (111) imply that

In turn $A_2 \le \frac{\delta }{16}.$ By Theorem 2.1 and (111) we get that $x_1$ and $x_2$ are Lebesgue’s points and

$$\begin{aligned} A_1+A_3=|{\varvec{\textsf {u}}}(x_1)-({\varvec{\textsf {u}}})_{B_{2r_\delta }(x_1)}|+|({\varvec{\textsf {u}}})_{B_{r_\delta }(x_2)}-{\varvec{\textsf {u}}}(x_2)|\le \frac{\delta }{4} . \end{aligned}$$

Applying these observation we get from (112) that

$$\begin{aligned} |{\varvec{\textsf {u}}}(x_1)-{\varvec{\textsf {u}}}(x_2)|&\le \frac{\delta }{2}. \end{aligned}$$

Since $x_2$ was an arbitrary point of $B_{r_\delta }(x_1)$, we have (110) justified, which completes the proof. $\square$

We are in the position to prove the Hölder continuity criterion.

Proof of Theorem 2.9

Notice that assumption (15) implies that there exists $c=c({{\texttt {\textit{data}}}})>0$, such that for all sufficiently small r we have

$$\begin{aligned} {{\mathcal {W}}}^{{\varvec{{\mathsf{\mu }}}}}_G(x,r)\le c r^\theta . \end{aligned}$$

Applying assumption (15) to Proposition 5.3 with $\alpha _V=\frac{\theta +1}{2}$, we have

for any $0< \rho < r \le R_0$. Now we apply Lemma 3.8 to see

By Campanato’s characterization [48, Theorem 2.9], we complete the proof. $\square$

Appendix

Proof of Lemma 2.5

Set $x\in \Omega _0,$ $R_k=2^{1-k}R$ and $B_k=B_{R_k}(x)$ for $k=0,1,\dots \,$. As ${\varvec{\textsf {F}}}$ is taken in a place of a measure with a slight abuse of notation we write $|{\varvec{\textsf {F}}}|(B_{R_k}(x))=\int _{B_k}|{\varvec{\textsf {F}}}(y)|\,dy.$ We notice that we have

$$\begin{aligned} {{\mathcal {W}}}_G^{|{\varvec{\textsf {F}}}|}(x,R)=\sum _{k=1}^\infty \int _{R_{k+1}}^{R_k} g^{-1}\left( \frac{|{\varvec{\textsf {F}}}|(B_r(x))}{r^{n-1}}\right) \,dr\lesssim \sum _{k=1}^\infty R_{k} g^{-1}\left( \frac{|{\varvec{\textsf {F}}}|(B_{R_k}(x))}{R_k^{n-1}}\right) \,. \end{aligned}$$

To estimate the series we employ the decreasing rearrangement $|{\varvec{\textsf {F}}}|^\star$ of $|{\varvec{\textsf {F}}}|$ and its maximal rearrangement $|{\varvec{\textsf {F}}}|^{\star \star }$. When $w_n$ is the volume of the unit ball, we have that

Then we have

$$\begin{aligned} R_kg^{-1}\left( \frac{|{\varvec{\textsf {F}}}|(B_{R_k}(x))}{R_k^{n-1}}\right)&\lesssim R_kg^{-1}\left( {w_n R_k}\, |{\varvec{\textsf {F}}}|^{\star \star }({w_nR_k^n})\right) \nonumber \\&\lesssim \int _{w_n R_k^{n}}^{w_n R_{k-1}^n}\rho ^\frac{1}{n}g^{-1}\left( \rho ^\frac{1}{n} \, |{\varvec{\textsf {F}}}|^{\star \star }(\rho )\right) \,\frac{d\rho }{\rho } \end{aligned}$$

(113)

with implicit constants independent of k. Therefore

$$\begin{aligned} \sup _{x\in \Omega _0}{{\mathcal {W}}}_G^{|{\varvec{\textsf {F}}}|}(x,R)&\lesssim \sum _{k=1}^\infty \int _{w_n R_k^{n}}^{w_n R_{k-1}^n} \rho ^{\frac{1}{n}-1} g^{-1}\left( \rho ^\frac{1}{n}\, |{\varvec{\textsf {F}}}|^{\star \star }(\rho )\right) \,\frac{d\rho }{\rho }\\&= \int _{0}^{w_n R^n} \rho ^\frac{1}{n} g^{-1}\left( \rho ^\frac{1}{n}\, |{\varvec{\textsf {F}}}|^{\star \star }(\rho )\right) \,\frac{d\rho }{\rho }\,. \end{aligned}$$

$\square$