Orlicz regularity of the gradient of solutions to quasilinear elliptic equations in the plane

Abstract

Given a planar domain Ω, we study the Dirichlet problem

$$\textstyle\begin{cases} {-}\operatorname {div}A(x, \nabla v)= f & \mbox{in } \Omega,\\ v=0 & \mbox{on } \partial\Omega, \end{cases} $$

where the higher-order term is a quasilinear elliptic operator, and f belongs to the Zygmund space $L (\log L)^{\delta} (\log\log\log L)^{\frac{\beta}{2}}(\Omega)$ with $\beta\geq0$ and $\delta\geq \frac{1}{2}$.

We prove that the gradient of the variational solution $v \in W^{1,2}_{0}(\Omega)$ belongs to the space $L^{2} (\log L)^{2\delta-1} (\log \log\log L)^{\beta}(\Omega)$.

1 Introduction

In this paper we consider the following Dirichlet problem on a bounded open set $\Omega\subset \mathbb {R}^{2}$ with $\mathcal{C}^{1}$ boundary:

$$ \textstyle\begin{cases} {-}\operatorname {div}A(x, \nabla v) = f & \mbox{in } \Omega,\\ v=0 & \mbox{on } \partial\Omega, \end{cases} $$

(1.1)

where f belongs to the Zygmund space $L(\log L)^{\delta} (\log\log\log L)^{\frac{\beta}{2}}(\Omega)$ with $\beta\geq0$ and $\delta\geq\frac{1}{2}$. We prove that the distributional gradient of the unique solution $v\in W^{1,2}_{0}(\Omega)$ to (1.1) satisfies $|\nabla v|\in L^{2} (\log L)^{2\delta-1} (\log\log\log L)^{\beta }(\Omega)$.

Here $A: \Omega\times \mathbb {R}^{2} \longrightarrow \mathbb {R}^{2}$ is a mapping of Leray-Lions type [1], that is,

$$ \begin{aligned} &A( \cdot, \xi) \mbox{ is measurable for all } \xi\in \mathbb {R}^{2}, \mbox{ and}\\ &A(x, \cdot) \mbox{ is continuous for almost every } x\in\Omega. \end{aligned} $$

(1.2)

Moreover, we assume that there exists $K\geq1$ such that, for almost every $x\in\Omega$ and for any $\xi, \eta\in \mathbb {R}^{2}$,

$$ \begin{aligned} (\mathrm{i}) &\quad \bigl|A(x, \xi) - A(x, \eta)\bigr| \leq K|\xi- \eta|,\\ (\mathrm{ii}) &\quad |\xi- \eta|^{2} \leq K \bigl\langle A(x, \xi) - A(x, \eta), \xi -\eta\bigr\rangle ,\\ (\mathrm{iii}) &\quad A(x,0)=0. \end{aligned} $$

(1.3)

In [2], under assumptions (1.2) and (1.3), the authors proved the existence and uniqueness of the solution to the Dirichlet problem with $f\in L^{1}(\Omega)$ in the grand Sobolev space $W^{1,2)}_{0}(\Omega)$. Precisely, $W^{1,2)}_{0}(\Omega)$ is the space of functions $v\in W^{1,1}_{0}(\Omega)$ whose gradients belong to the grand Lebesgue space $L^{2)}(\Omega)$ (see Section 2 for a definition).

Nowadays, a vast literature is available dealing with several types of a priori estimates on the gradients of solutions to equations of this kind; see, for example, [3–5].

We are interested in cases where the solution is the variational $W^{1,2}(\Omega)$ solution. The minimal assumption on f that guarantees this is $f\in L (\log L)^{\frac{1}{2}}(\Omega)$. This follows by the embedding in the plane (see [6, 7], and [8])

$$W^{1, 2}_{0} (\Omega) \hookrightarrow\exp_{2}( \Omega) $$

and by the duality relation (see [9])

$$\bigl((\exp_{2}) (\Omega) \bigr)'=L(\log L)^{\frac{1}{2}} (\Omega) . $$

In [10], the authors interpolate between the data spaces

$$L(\log L)^{\frac{1}{2}} (\Omega) \quad\mbox{and}\quad L(\log L) (\Omega). $$

To this aim, the following estimate was proved for $0 \leq\beta\leq1$:

$$ \| \nabla v\|_{L^{2} (\log L)^{\beta}(\Omega)} \leq C(K, \beta) \| f \|_{L (\log L)^{\frac{(\beta+1)}{2}}(\Omega)}. $$

(1.4)

When f belongs to the Zygmund space $L(\log L)^{\frac{1}{2}} (\log \log L)^{\frac{\beta}{2}}(\Omega)$ for $0\leq\beta<2$, the unique solution v to the Dirichlet problem (1.1) satisfies $|\nabla v|\in L^{2}(\log\log L)^{\beta}(\Omega)$ with the estimate

$$ \| \nabla v\|_{L^{2} (\log\log L)^{\beta}(\Omega)} \leq C(K, \beta) \| f\|_{L (\log L)^{\frac{1}{2}} (\log\log L)^{\frac{\beta}{2}}(\Omega)} $$

(1.5)

(see [11]). This generalizes a result of [12] obtained for $\beta=1$.

Starting from the results of [11], in [13], the authors of the present paper prove an analogue of the previous result when the critical Zygmund class $L(\log L)^{\frac{1}{2}}(\Omega)$ is perturbed in a weaker way, namely with perturbations of order $\log \log \log L$. Precisely, in [13], it is proved that if $\beta\geq 0$, then

$$ \| \nabla v\|_{L^{2} (\log\log\log L)^{\beta}(\Omega)} \leq C(K, \beta) \| f \|_{L (\log L)^{\frac{1}{2}} (\log\log\log L)^{\frac{\beta }{2}}(\Omega)}. $$

(1.6)

The aim of this paper is to extend the results of [13] to the case $f\in L(\log L)^{\delta} (\log\log \log L)^{\frac {\beta}{2}}(\Omega)$ with $\beta\geq0$ and $\delta\geq\frac{1}{2}$, that is, to prove the following:

Theorem 1.1

Let $A= A(x, \xi)$ satisfy (1.2) and (1.3), and let $\beta \geq0$, $\delta\geq\frac{1}{2}$. Then, if $f\in L (\log L)^{\delta }(\log \log\log L)^{\frac{\beta}{2}}(\Omega)$, the gradient of the unique finite energy solution $v\in W^{1,2}_{0}(\Omega)$ to the Dirichlet problem (1.1) belongs to the Orlicz space $L^{2} (\log L)^{2\delta -1} (\log\log \log L)^{\beta}(\Omega, \mathbb {R}^{2})$, and the following estimate holds:

$$\| \nabla v\|_{L^{2} (\log L)^{2\delta-1}(\log\log\log L)^{\beta }(\Omega ;\mathbb {R}^{2})} \leq C(K, \beta, \delta) \| f\|_{L (\log L)^{\delta} (\log \log\log L)^{\frac{\beta}{2}}(\Omega)}. $$

In order to prove this theorem, we will find an integral expression equivalent to the Luxemburg norm in the Zygmund class (see Theorem 3.1), which is based on a method recently introduced in [14, 15].

We note that our method allows us to prove estimates (1.4) and (1.6) for any $\beta\geq0$ (in particular, see Lemmas 2.3 and 2.4).

2 Preliminaries

Let Ω be a bounded domain in $\mathbb {R}^{n}$, $n\geq2$. A function u belongs to the Lebesgue space $L^{p}(\Omega)$ with $1\leq p < \infty$ if and only if

$$\| u\|_{L^{p}(\Omega)}= \biggl( \fint_{\Omega}|u|^{p} \,dx \biggr)^{\frac {1}{p}}< + \infty, $$

where $\fint_{\Omega}=\frac{1}{|\Omega|} \int_{\Omega}$.

Now we recall some useful function spaces slightly larger than the classical Lebesgue spaces.

2.1 Grand Lebesgue spaces

For $1< p<\infty$, let us consider the class, denoted by $L^{p)}(\Omega )$, consisting of all measurable functions $u\in\bigcap_{1\leq q < p} L^{q}(\Omega)$ such that

$$\sup_{0< \varepsilon\leq p-1} \biggl\{ \varepsilon \fint_{\Omega} \bigl|u(x)\bigr|^{p- \varepsilon} \biggr\} ^{\frac{1}{p-\varepsilon}} < + \infty $$

which was introduced in [16]; $L^{p)}(\Omega)$ becomes a Banach space, the grand Lebesgue space $L^{p)}(\Omega)$, equipped with the norm

$$\| u \|_{L^{p)}(\Omega)}= \sup_{0< \varepsilon\leq p-1} \varepsilon ^{\frac{1}{p}} \biggl\{ \fint_{\Omega} \bigl|u(x)\bigr|^{p- \varepsilon} \biggr\} ^{\frac{1}{p-\varepsilon}}. $$

Moreover, $\| u \|_{L^{p)}(\Omega)}$ is equivalent to

$$\sup_{0< \varepsilon\leq p-1} \biggl\{ \varepsilon \fint_{\Omega} \bigl|u(x)\bigr|^{p- \varepsilon} \biggr\} ^{\frac{1}{p-\varepsilon}}. $$

In general, if $0<\alpha<\infty$, then we can define the space $L^{\alpha, p)}(\Omega)$ as the space of all measurable functions $u\in \bigcap_{1\leq q < p} L^{q}(\Omega)$ such that

$$\|u\|_{L^{\alpha, p)}(\Omega)}=\sup_{0< \varepsilon\leq p-1} \bigl\{ \varepsilon^{\frac{\alpha}{p}}\|u\|_{p- \varepsilon} \bigr\} < + \infty . $$

2.2 Orlicz spaces

Let Ω be an open set in $\mathbb {R}^{n}$ with $n\geq2$. A function $\Phi: [0, + \infty) \rightarrow[0, + \infty)$ is called a Young function if it is convex, left-continuous, and vanishes at 0; thus, any Young function Φ admits the representation

$$\Phi(t)= \int_{0}^{t} \phi(s)\,ds \quad\mbox{for } t \geq0, $$

where $\phi: [0, + \infty) \rightarrow[0, + \infty)$ is a nondecreasing left-continuous function that is neither identically equal to 0 nor to ∞.

The Orlicz space associated to Φ, named $L^{\Phi }(\Omega)$, consists of all Lebesgue-measurable functions $f: \Omega \rightarrow \mathbb {R}$ such that

$$\int_{\Omega} \Phi\bigl(\lambda|f|\bigr) < \infty\quad\mbox{for some } \lambda= \lambda(f)>0. $$

$L^{\Phi}(\Omega)$ is a Banach space equipped with the Luxemburg norm

$$\| f \|_{L^{\Phi}(\Omega)} = \inf \biggl\{ \frac{1}{\lambda}: \int _{\Omega} \Phi\bigl(\lambda|f|\bigr) \leq1 \biggr\} . $$

Examples of Orlicz spaces:

• (1) If $\Phi(t)= t^{p}$ for $1 \leq p < \infty$, then $L^{\Phi }(\Omega)$ is the classical Lebesgue space $L^{p}(\Omega)$.

• (2) If $\Phi(t)= t^{p} ( \log(a+ t) )^{q}$ with either $p>1$ and $q\in \mathbb {R}$ or $p=1$ and $q\geq0$ and where $a \geq e$, then $L^{\Phi}(\Omega)$ is the Zygmund space denoted by $L^{p} (\log L)^{q}(\Omega)$.

• (3) If $\Phi(t)= t^{p} (\log(a+t) )^{q_{1}} ( \log\log \log(a + t) )^{q_{2}}$ with either $p>1$ and $q_{1},q_{2}\in \mathbb {R}$ or $p=1$ and $q_{1},q_{2}\geq0$ and where $a \geq e^{e^{e}}$, then $L^{\Phi }(\Omega)$ is the space $L^{p} (\log L)^{q_{1}}(\log\log\log L)^{q_{2}} (\Omega)$.

• (4) If $\Phi(s)= e^{t^{a}}-1$ and $a>0$, then $L^{\Phi}(\Omega )$ is the space of a-exponentially integrable functions $\operatorname {EXP}_{a}(\Omega)$.

  We denote by $\exp_{a}(\Omega)$ the closure of $L^{\infty}(\Omega )$ in $\operatorname {EXP}_{a}(\Omega)$.

The Young complementary function is given by

$$\tilde{\Phi}(t)= \int_{0}^{t} \phi^{-1}(s)\,ds, $$

where

$$\phi^{-1}(s)= \sup\bigl\{ r: \phi(r) \leq s \bigr\} . $$

Moreover, the following Hölder-type inequality holds:

$$\biggl\vert \int_{\Omega} f(x) g(x)\,dx \biggr\vert \leq C(\Phi) \| f\| _{L^{\Phi}(\Omega)}\| g\|_{L^{\tilde{\Phi}}(\Omega)} $$

for $f\in L^{\Phi}(\Omega)$ and $g\in L^{\tilde{\Phi }}(\Omega)$.

Definition 2.1

A Young function Φ satisfies the $\Delta_{2}$-condition ($\Phi\in \Delta_{2}$) if

$$\Phi(2s) \leq C \Phi(s) $$

for some constant $C\geq2$ and all $s>0$.

By the Riesz representation theorem, if Φ and Φ̃ belong to the class $\Delta_{2}$, then the dual space of $L^{\Phi }(\Omega)$ is $L^{\tilde{\Phi}}(\Omega)$.

Now we recall the explicit expression of the duals of some Orlicz spaces (see [17–19]).

Theorem 2.1

Let $\Omega\subset \mathbb {R}^{n}$ be an open set. If $1 < p < \infty$ and $q, q_{1}, q_{2}\in \mathbb {R}$, then

• $(L^{p} (\log L)^{q}(\Omega) )' \cong L^{p'} (\log L)^{-\frac{q}{p-1}}(\Omega)$,

• $(L^{p} (\log\log\log L)^{q} (\Omega) )' \cong L^{p'} (\log\log\log L)^{-\frac{q}{p-1}}(\Omega)$,

• $(L^{p} (\log L)^{q_{1}}(\log\log\log L)^{q_{2}} (\Omega ) )' \cong L^{p'} (\log L)^{-\frac{q_{1}}{p-1}}(\log\log\log L)^{-\frac {q_{2}}{p-1}}(\Omega)$,

where $p'$ is the conjugate exponent of p, that is, $\frac{1}{p}+ \frac{1}{p'}=1$.

If $p=1$ and $q>0$, then

• $(L(\log L)^{q}(\Omega) )' \cong \operatorname {EXP}_{\frac{1}{q}} (\Omega)$.

Given two Young functions Φ and Ψ, we say that Ψ dominates Φ globally (respectively near infinity) if there exists a constant $k>0$ such that

$$\Phi(t)\leq\Psi(kt) \quad\mbox{for all } t\geq0 \mbox{ (respectively for all } t \geq t_{0} \mbox{ for some } t_{0}>0) ; $$

moreover, Φ and Ψ are equivalent globally (respectively near infinity, $\Phi\cong\Psi$) if each dominates the other globally (respectively near infinity). If Φ̃ and Ψ̃ are the complementary Young functions of, respectively, Φ and Ψ, then Ψ dominates Φ globally (or near infinity) if and only if Φ̃ dominates Ψ̃ globally (or near infinity). Similarly, Φ and Ψ are equivalent if and only if Φ̃ and Ψ̃ are equivalent. We have the following result.

Theorem 2.2

The continuous embedding $L^{\Psi}(\Omega)\hookrightarrow L^{\Phi }(\Omega)$ holds if and only if either Ψ dominates Φ globally or Ψ dominates Φ near infinity and Ω has finite measure.

Finally, we recall the definition of the Orlicz-Sobolev spaces $W^{1,\Psi}(\Omega)$ and $W^{1,\Psi }_{0}(\Omega)$ (see [20–23]). The space $W^{1,\Psi }(\Omega)$ consists of the equivalence classes of functions u in $L^{\Psi}(\Omega)$ whose distributional gradients ∇u belong to $L^{\Psi}$. This is a Banach space with respect to the norm given by

$$\|u\|_{W^{1,\Psi}(\Omega)}=\|u\|_{L^{\Psi}(\Omega)}+\|\nabla u\| _{L^{\Psi}(\Omega)} . $$

As in the case of the ordinary Sobolev space, $W^{1,\Psi }_{0}(\Omega)$ coincides with the closure of $C^{\infty}_{0}(\Omega)$ in $W^{1,\Psi}(\Omega)$.

2.3 Orlicz-Sobolev imbeddings

Lemma 2.3

Let $\Phi(t)=\exp \{\frac{t^{\frac{1}{\delta}}}{ (\log (e + \log (e+ t)) )^{\frac{\beta}{2\delta}}} \}-1$ with $\beta\in \mathbb {R}$ and $\delta>0$. Then

$$ \tilde{\Phi}(t)\cong t (\log t )^{\delta} ( \log \log \log t )^{\frac{\beta}{2}}. $$

(2.1)

Proof

Since Φ is a Young function, by definition we have

$$\Phi(t)= \int_{0}^{t} \phi(s)\,ds, $$

where ϕ is equivalent near infinity to

$$\Phi(s) \cdot \biggl[\frac{s^{\frac{1}{\delta}-1}}{\delta (\log\log s )^{\frac{\beta}{2\delta}}}- \frac{\beta s^{\frac{1}{\delta }-1}}{2\delta(\log s) \cdot (\log\log s )^{\frac{\beta }{2\delta } +1}} \biggr]. $$

For large s, we have

$$\phi(s)\cong\Phi(s) \frac{s^{\frac{1}{\delta}-1}}{\delta (\log\log s )^{\frac{\beta}{2\delta}}}, $$

and we will prove that, near infinity,

$$ \phi(s)\cong\Phi(s). $$

(2.2)

We begin with the case $\delta\leq1$. Then we can state that there exists $c>1$ such that

$$\begin{aligned} \exp \biggl\{ \frac{s^{\frac{1}{\delta}}}{ (\log\log s )^{\frac {\beta}{2\delta}} } \biggr\} &\leq\exp \biggl\{ \frac{s^{\frac {1}{\delta }}}{ (\log\log s )^{\frac{\beta}{2\delta}} } \biggr\} \cdot \frac{s^{\frac{1}{\delta}-1}}{\delta (\log\log s )^{\frac{\beta }{2\delta}}} \\ &\leq\exp \biggl\{ \frac{(cs)^{\frac{1}{\delta }}}{ (\log\log(cs) )^{\frac{\beta}{2\delta}} } \biggr\} . \end{aligned}$$

Similarly, in the case $\delta>1$, there exists $c\in(0,1)$ such that

$$\begin{aligned} \exp \biggl\{ \frac{(cs)^{\frac{1}{\delta}}}{ (\log\log (cs) )^{\frac{\beta}{2\delta}} } \biggr\} &\leq\exp \biggl\{ \frac {s^{\frac {1}{\delta}}}{ (\log\log s )^{\frac{\beta}{2\delta}} } \biggr\} \cdot \frac{s^{\frac{1}{\delta}-1}}{\delta (\log\log s )^{\frac{\beta }{2\delta}}}\\ &\leq\exp \biggl\{ \frac{s^{\frac{1}{\delta }}}{ (\log \log s )^{\frac{\beta}{2\delta}} } \biggr\} . \end{aligned}$$

Hence, (2.2) is proved, and then it is not difficult to check that

$$\phi^{-1}(r) \cong(\log r)^{\delta} (\log\log\log r )^{\frac {\beta}{2}}. $$

By the definition of a complementary Young function, for large y, we obtain that

$$\tilde{\Phi}(y)= \int_{0}^{y} \phi^{-1} (r)\,dr\cong y ( \log y)^{\delta } (\log\log\log y )^{\frac{\beta}{2}}. $$

 □

Given a Young function Ψ such that

$$\int_{0} \biggl(\frac{r}{\Psi(r)} \biggr)\,dr< \infty, $$

we define $\Phi: [0, + \infty) \rightarrow[0, + \infty)$ as

$$ \Phi(s)= \Psi\circ H^{-1}_{2}(s) \quad\mbox{for } s \geq0 , $$

(2.3)

where $H^{-1}_{2}(s)$ is the (generalized) left-continuous inverse of the function $H_{2}: [0, + \infty) \rightarrow[0, + \infty)$ given by

$$ H_{2}(r)= \biggl( \int_{0}^{r} \biggl(\frac{t}{\Psi(t)} \biggr)\,dt \biggr)^{\frac{1}{2}} \quad\mbox{for } r \geq0 . $$

(2.4)

In [24] and in [25], the author showed that Φ is a Young function and that the following Sobolev-Orlicz embedding theorem holds:

$$\| u\|_{L^{\Phi}(\Omega)} \leq C \| \nabla u \|_{L^{\Psi}(\Omega)} $$

for every function u in the Orlicz-Sobolev space $W^{1, \Psi}(\Omega )$. As an application, we prove an embedding theorem, which can be regarded as an extension of Lemma 2.4 in [13].

Lemma 2.4

Let $\Omega\subset \mathbb {R}^{2}$ be an open bounded set with $\mathcal{C}^{1}$ boundary. Consider the Young function

$$\Psi(t)= t^{2} (\log t )^{1-2\delta} ( \log\log\log t )^{-\beta} $$

with $\beta\in \mathbb {R}$ and ${\delta\geq\frac{1}{2}}$. Then

$$W^{1, \Psi}(\Omega) \hookrightarrow L^{\Phi}(\Omega), $$

where

$$ \Phi(s)\cong e^{s^{\frac{1}{\delta}} ( \log\log s )^{-\frac {\beta}{2\delta}}} . $$

(2.5)

Proof

By (2.4) we have that

$$H_{2}(r)= \biggl( \int_{0}^{r} \frac{(\log t)^{2\delta-1}(\log\log\log t)^{\beta}}{t}\,dt \biggr)^{\frac{1}{2}} \cong ( \log r )^{\delta} (\log\log\log r)^{\frac{\beta}{2}}. $$

Moreover, as shown in the proof of Lemma 2.3, the inverse function $H^{-1}_{2}(s)$ is equivalent near infinity to

$$e^{s^{\frac{1}{\delta}} ( \log\log s )^{-\frac{\beta }{2\delta}}}. $$

By (2.3) we obtain that

$$\begin{aligned} \Phi(s) &\cong e^{2s^{\frac{1}{\delta}} ( \log\log s )^{-\frac{\beta}{2\delta}}} \bigl(s^{\frac{1}{\delta}}( \log\log s)^{-\frac {\beta}{2\delta}} \bigr)^{1-2\delta} \bigl(\log\log s^{\frac {1}{\delta }}( \log\log s)^{-\frac{\beta}{2\delta}} \bigr)^{-\beta}\\ &\cong e^{s^{\frac{1}{\delta}} ( \log\log s )^{-\frac{\beta }{2\delta}}}, \end{aligned}$$

and we conclude that

$$W^{1, \Psi}(\Omega) \hookrightarrow L^{\Phi} (\Omega) . $$

 □

Remark 2.5

The previous lemma for $\delta=\frac{1}{2}$ and $\beta=0$ was proved in [6, 7], and [8]. The case $\beta=0$ and $\delta> \frac{1}{2}$ is proved in [26].

3 Equivalent norm on the Zygmund spaces $L^{q}(\log L)^{-\gamma }(\log\log\log L)^{-\beta}(\Omega)$

The main tool of this section is to obtain an integral expression equivalent to the Luxemburg norm in $L^{q}(\log L)^{-\gamma} ( \log \log\log L )^{-\beta}(\Omega)$ with $1< q<\infty$, $\beta\geq0$ and $\gamma>0$.

If f is a measurable function on Ω, we set

$$ |\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} ( \log\log\log L )^{-\beta }(\Omega)}= \biggl\{ \int_{0}^{\varepsilon_{0}} \varepsilon^{\gamma-1} \| f\| _{L^{q- \varepsilon} ( \log\log\log L )^{-\beta}(\Omega )}^{q} \,d\varepsilon \biggr\} ^{\frac{1}{q}} . $$

(3.1)

Here $\varepsilon_{0} \in \,] 0, q-1]$ is fixed.

For $\beta=0$, (3.1) becomes

$$|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma}(\Omega)}= \biggl\{ \int _{0}^{\varepsilon_{0}} \varepsilon^{\gamma-1} \| f \|_{L^{q- \varepsilon}(\Omega)}^{q} \,d\varepsilon \biggr\} ^{\frac{1}{q}} $$

as in [15].

Theorem 3.1

We have $f\in L^{q} (\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)$ if and only if

$$|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)} < + \infty. $$

Moreover, $|\!|\!|\cdot |\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta}(\Omega)}$ is a norm equivalent to the Luxemburg one, that is, there exist constants $C_{i}= C_{i}(q, \beta,\gamma,\varepsilon _{0})$, $i=1, 2$, such that, for all $f\in L^{q}(\log L)^{-\gamma} (\log \log \log L )^{-\beta}(\Omega)$,

$$\begin{aligned} C_{1} \| f \|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)} &\leq|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log \log\log L )^{-\beta}(\Omega)}\\ &\leq C_{2} \| f \|_{L^{q}(\log L)^{-\gamma } (\log\log\log L )^{-\beta}(\Omega)}. \end{aligned}$$

Proof

It is easy to check that $|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log \log L )^{-\beta}(\Omega)}$, defined by (3.1), is a norm on $L^{q}(\log L)^{-\gamma}(\log\log\log L)^{-\beta}(\Omega)$.

Moreover, for any measurable function f and for a.e. $x\in\Omega$, if $a\geq e^{e^{e}}$, then we have

$$|f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\leq|f|^{q-\varepsilon} \leq2^{q-1} \bigl[a^{q}+|f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon} \bigr], $$

and so we deduce

$$\begin{aligned} |f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\bigl(\log\log\log\bigl(a+|f|\bigr) \bigr)^{-\beta}\leq{}& |f|^{q-\varepsilon}\bigl(\log\log\log\bigl(a+|f|\bigr) \bigr)^{-\beta}\\ \leq{}&2^{q-1} \bigl[a^{q}+|f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon} \bigr]\\ &{}\times\bigl(\log\log \log \bigl(a+|f|\bigr)\bigr)^{-\beta} . \end{aligned}$$

Integrating over Ω, we get

$$\begin{aligned} &\fint_{\Omega} |f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\bigl(\log \log\log \bigl(a+|f|\bigr)\bigr)^{-\beta}\,dx\\ &\quad\leq\|f\|_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega)} ^{q-\varepsilon} \\ &\quad\leq2^{q-1}a^{q} +2^{q-1} \fint_{\Omega}|f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon }\bigl(\log \log\log\bigl(a+|f|\bigr)\bigr)^{-\beta}\,dx . \end{aligned}$$

Then we multiply for $\varepsilon^{\gamma-1}$ and integrate between 0 and $\varepsilon_{0}$ to obtain:

$$\begin{aligned} & \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma-1} \biggl[ \fint _{\Omega} |f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\bigl(\log \log\log\bigl(a+|f|\bigr)\bigr)^{-\beta}\,dx \biggr]\,d\varepsilon \\ &\quad \leq \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma -1}\|f\|_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega)} ^{q-\varepsilon}\,d\varepsilon\\ &\quad\leq2^{q-1}a^{q} \frac{\varepsilon_{0}^{\gamma}}{\gamma}+ 2^{q-1} \int _{0}^{\varepsilon_{0}}\varepsilon^{\gamma-1} \biggl[ \fint_{\Omega} |f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\bigl(\log \log\log\bigl(a+|f|\bigr)\bigr)^{-\beta}\,dx \biggr]\,d\varepsilon. \end{aligned}$$

Thanks to Lemma 4.3 of [11], used with the choice $b=a+|f|$, we obtain that there exist two constant $C_{1}$, $C_{2}$, depending only on γ and $\varepsilon_{0}$, such that

$$\begin{aligned} C_{1} \fint_{\Omega} & |f|^{q}\bigl(\log\bigl(a+|f|\bigr) \bigr)^{-\gamma} \bigl(\log \log\log \bigl(a+|f|\bigr) \bigr)^{-\beta}\,dx \\ & \leq \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma -1}\|f\|_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega )}^{q-\varepsilon}\,d\varepsilon \\ & \leq C_{2} \biggl[1+ \fint_{\Omega} |f|^{q}\bigl(\log \bigl(a+|f|\bigr) \bigr)^{-\gamma} \bigl(\log\log\log\bigl(a+|f|\bigr) \bigr)^{-\beta}\,dx \biggr]. \end{aligned}$$

(3.2)

If $|\!|\!|f|\!|\!|_{L^{q} (\log\log\log L )^{-\beta}(\Omega )}$ is finite, then since

$$ \|f\|^{q-\varepsilon} _{L^{q-\varepsilon}(\log\log\log L)^{-\beta }(\Omega)}\leq\|f\|^{q} _{L^{q-\varepsilon}(\log\log\log L)^{-\beta }(\Omega)}+1, $$

by the first inequality in (3.2) we get that $f\in L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta}(\Omega)$. Moreover, if $|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log \log L )^{-\beta}(\Omega)}=1$, then

$$\fint_{\Omega} |f|^{q}\bigl(\log(a+|f|) \bigr)^{-\gamma} \bigl(\log\log\log (a+|f|) \bigr)^{-\beta}\,dx \leq C_{3} , $$

where $C_{3}$ is a constant independent on f. By homogeneity, for any measurable f, we get

$$\|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega )}\leq C_{3} |\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log \log L )^{-\beta}(\Omega)} . $$

Before proving the converse, we recall that

$$ \sup_{0 < \sigma\leq q-1} \sigma^{\frac{\gamma}{q-\sigma}} \| f\| _{L^{q-\sigma} (\log\log\log L )^{-\beta}(\Omega)} \leq C_{4} \| f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)}. $$

(3.3)

Indeed, if we fix $a\geq e^{e^{e}}$ and proceed as in Lemma 1.2 in [16], using the Hölder inequality and the inequality

$$\log^{\lambda}(a+t)\leq\lambda^{\lambda}(a+t), $$

we obtain

$$\begin{aligned} &\int_{\Omega}|f|^{q-\sigma} \bigl(\log\log\log \bigl(a+|f| \bigr) \bigr)^{-\beta}\\ &\quad= \int_{\Omega}\frac{|f|^{q-\sigma} (\log\log\log (a+|f|) )^{-\beta+\frac{\beta(q-\sigma)}{q}-\frac{\beta(q-\sigma )}{q}} (\log (a+|f|) )^{\frac{\gamma(q-\sigma)}{q}}}{ (\log(a+|f|) )^{\frac{\gamma(q-\sigma)}{q}}}\\ &\quad\leq \biggl[ \int_{\Omega}\frac{|f|^{q} (\log\log\log (a+|f|) )^{-\beta}}{ (\log(a+|f|) )^{\gamma}} \biggr]^{\frac {q-\sigma }{q}} \\ &\qquad{}\times\biggl[ \int_{\Omega}\bigl(\log\log\log \bigl(a+|f| \bigr) \bigr)^{(-\beta+\frac {\beta(q-\sigma)}{q})\frac{q}{\sigma}} \bigl(\log\bigl(a+|f|\bigr) \bigr)^{\frac {\gamma(q-\sigma)}{\sigma}} \biggr]^{\frac{\sigma}{q}} \\ &\quad\leq \biggl[ \int_{\Omega}\frac{|f|^{q} (\log\log\log (a+|f|) )^{-\beta}}{ (\log(a+|f|) )^{\gamma}} \biggr]^{\frac {q-\sigma }{q}} \\ &\qquad{}\times\biggl[ \biggl(\frac{\gamma(q-\sigma)}{\sigma} \biggr)^{\frac {\gamma (q-\sigma)}{\sigma}} \int_{\Omega}\bigl(\log\log\log \bigl(a+|f| \bigr) \bigr)^{-\beta}\bigl(a+|f|\bigr) \biggr]^{\frac{\sigma}{q}}\\ &\quad\leq \biggl[ \int_{\Omega}\frac{|f|^{q} (\log\log\log (a+|f| ) )^{-\beta}}{ (\log(a+|f|) )^{\gamma}} \biggr]^{\frac {q-\sigma }{q}} \biggl[ \biggl(\frac{\gamma(q-\sigma)}{\sigma} \biggr)^{\frac {\gamma (q-\sigma)}{\sigma}} \int_{\Omega}\bigl(a+|f|\bigr) \biggr]^{\frac{\sigma}{q}}. \end{aligned}$$

Hence, elevating both sides of this inequality to the power $\frac {1}{q-\sigma}$ and then multiplying both of them by $\sigma^{\frac {\gamma}{q-\sigma}}$, we deduce

$$\begin{aligned} \biggl[\sigma^{\gamma}&\int_{\Omega}|f|^{q-\sigma} \bigl(\log\log\log \bigl(a+|f| \bigr) \bigr)^{-\beta} \biggr]^{\frac{1}{q-\sigma}}\\ &\quad\leq \biggl[ \int_{\Omega}\frac{|f|^{q} (\log\log\log (a+|f| ) )^{-\beta}}{ (\log(a+|f|) )^{\gamma}} \biggr]^{\frac {1}{q}} \bigl(a|\Omega|+ \|f\|_{L^{1}(\Omega)} \bigr)^{\frac{\sigma}{q(q-\sigma )}}\gamma ^{\frac{\gamma}{q}}(q-\sigma)^{\frac{\gamma}{q}}\sigma^{\frac {\gamma \sigma}{q(q-\sigma)}}, \end{aligned}$$

and passing to the supremum with respect to $\sigma\in(0,q-1]$, we get formula (3.3) with

$$C_{4}=\gamma^{\frac{\gamma}{q}}\sup_{0< \sigma\leq q-1} \bigl\{ \bigl(a| \Omega|+\|f\|_{L^{1}(\Omega)} \bigr)^{\frac{\sigma}{q(q-\sigma )}}(q- \sigma)^{\frac{\gamma}{q}}\sigma^{\frac{\gamma\sigma }{q(q-\sigma )}} \bigr\} . $$

If $f\in L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)$, that is, if

$$ \|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega )}< \infty $$

(3.4)

by (3.3), then there exists a constant $C_{5}$ independent on f such that

$$ \|f\|_{L^{q-\varepsilon} (\log\log\log L )^{-\beta }(\Omega )}\leq C_{5} \varepsilon^{-\frac{\gamma}{q-\varepsilon}} \|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)}. $$

(3.5)

By (3.5) we get

$$\begin{aligned} \|f\|^{q}_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega)}&= \|f\|^{q-\varepsilon}_{L^{q-\varepsilon}(\log\log\log L)^{-\beta }(\Omega )} \|f\|^{\varepsilon}_{L^{q-\varepsilon}(\log\log\log L)^{-\beta }(\Omega )} \\ &\leq C_{6} \|f\|^{q-\varepsilon}_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega)} \|f\|^{\varepsilon}_{L^{q}(\log L)^{-\gamma } (\log\log\log L )^{-\beta}(\Omega)}. \end{aligned}$$

(3.6)

Hence, by (3.2) we obtain that $|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma } (\log\log\log L )^{-\beta}(\Omega)} <+\infty$. Indeed, if

$$\|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega )}=1, $$

by (3.6) and (3.2) we get

$$|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)}< C_{7} , $$

where the constant $C_{7}$ is independent on f. By homogeneity we conclude the proof, obtaining

$$|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)}< C_{7} \|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta}(\Omega)} . $$

 □

4 Proof of Theorem 1.1

In this section, before proving Theorem 1.1, we state a regularity result for elliptic equations with right-hand side in divergence form. For convenience of the reader, we recall Theorem 3.1 of [2].

Theorem 4.1

Let $A=A(x,\xi)$ be a Leray-Lions mapping that satisfies (1.3). Then there exists $\sigma_{0}= \sigma_{0}(K)>0$ such that, for $|\sigma |\leq\sigma_{0}$ and $\underline{\chi}_{1}$, $\underline{\chi}_{2}\in L^{2-\sigma}( \Omega; \mathbb {R}^{2})$, each of the two problems

$$\begin{aligned}& \textstyle\begin{cases} \operatorname {div}A(x,\nabla\varphi_{1})=\operatorname {div}\underline{\chi}_{1} & \textit{in } \Omega,\\ \varphi_{1}\in W^{1,2-\sigma}_{0}(\Omega) , \end{cases}\displaystyle \end{aligned}$$

(4.1)

$$\begin{aligned}& \textstyle\begin{cases} \operatorname {div}A(x,\nabla\varphi_{2})=\operatorname {div}\underline{\chi}_{2} & \textit{in } \Omega,\\ \varphi_{2}\in W^{1,2-\sigma}_{0}(\Omega) , \end{cases}\displaystyle \end{aligned}$$

(4.2)

has a unique solution and

$$\| \nabla\varphi_{1}-\nabla\varphi_{2} \|_{L^{2-\sigma}(\Omega)} \leq C(K) \| \underline{\chi}_{1} - \underline{\chi}_{2} \|_{L^{2-\sigma }(\Omega)}, $$

where $C(K)>0$ depends only on K.

Theorem 4.1 allows us to prove the following:

Theorem 4.2

Let $A=A(x,\xi)$ be a Leray-Lions mapping that satisfies (1.3). Then, if $\gamma>0$ and $\beta\geq0$, for $i=1,2$ and for any $\underline{\chi}_{i}\in L^{2}(\log L)^{-\gamma}(\log\log\log L)^{-\beta }(\Omega; \mathbb {R}^{2})$, there exists a unique solution $\varphi_{i}$ to the Dirichlet problem

$$\begin{aligned} \textstyle\begin{cases} \operatorname {div}A(x,\nabla\varphi_{i})=\operatorname {div}\underline{\chi}_{i} & \textit{in } \Omega,\\ \varphi_{i}\in W^{1,1}_{0}(\Omega) . \end{cases}\displaystyle \end{aligned}$$

(4.3)

Moreover,

$$ \|\nabla\varphi_{1}-\nabla\varphi_{2} \|_{L^{2}(\log L)^{-\gamma}(\log \log\log L)^{-\beta}(\Omega)}\leq C\|\underline{\chi}_{1}-\underline{\chi }_{2}\| _{L^{2}(\log L)^{-\gamma}(\log\log\log L)^{-\beta}(\Omega)}, $$

(4.4)

where $C=C(\beta,\gamma,K)>0$ is a positive constant that depends on the parameters K, β, and γ.

Proof

By Theorem 4.1 there exists a positive constant $\sigma_{0}=\sigma (K)$ such that if $|\sigma|\leq\sigma_{0}$, then for $i=1,2$ and for any $\underline{\chi}_{i}\in L^{2-\sigma}(\Omega; \mathbb {R}^{2})$, problem (4.3) admits a unique solution $\varphi_{i} \in W^{1, 2-\sigma}_{0}$, and

$$ \|\nabla\varphi_{1}-\nabla\varphi_{2} \|_{L^{2-\sigma}(\Omega)}\leq C\| \underline{\chi}_{1}-\underline{ \chi}_{2}\|_{L^{2-\sigma}(\Omega)} , $$

(4.5)

where $C=C(K)>0$ is a positive constant that depends only on the parameter K.

If $\gamma>0$ and $\beta\geq0$ are fixed, using Theorem 3.1, we obtain

$$\begin{aligned} &\|\nabla\varphi_{1}-\nabla \varphi_{2}\|^{2}_{L^{2}(\log L)^{-\gamma}(\log \log \log L)^{-\beta}(\Omega)} \\ &\quad\leq C_{1}(\beta, \gamma)|\!|\!|\nabla\varphi_{1}-\nabla\varphi _{2} |\!|\!|^{2}_{L^{2}(\log L)^{-\gamma}(\log\log\log L)^{-\beta}(\Omega )}\\ &\quad=C_{1}(\beta,\gamma) \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma-1} \| \nabla \varphi_{1}-\nabla\varphi_{2}\|_{L^{2- \varepsilon} ( \log \log \log L )^{-\beta}(\Omega)}^{2}\,d\varepsilon. \end{aligned}$$

For $\beta=0$, by Theorem 4.1 we get

$$ \|\nabla\varphi_{1}-\nabla\varphi_{2}\|^{2}_{L^{2}(\log L)^{-\gamma }(\Omega )} \leq C_{2}(\gamma, K) \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma -1} \| \underline{\chi}_{1}-\underline{\chi}_{2}\|_{L^{2- \varepsilon}(\Omega)}^{2}\,d\varepsilon. $$

If $\beta>0$, then with a suitable choice of $\lambda_{0}$, by Theorem 3 in [13] and Theorem 4.1, we get

$$\begin{aligned} &\|\nabla\varphi_{1}-\nabla \varphi_{2}\|^{2}_{L^{2}(\log L)^{-\gamma}(\log \log \log L)^{-\beta}(\Omega)}\\ &\quad\leq C_{3}(\beta,\gamma) \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma-1} \biggl[ \int_{0}^{\lambda_{0}}\bigl(1+\log|\log\lambda|\bigr)^{-\beta -1}\bigl(\lambda|\log \lambda|\bigr)^{-1}\\ &\qquad{}\times\| \nabla\varphi_{1}-\nabla \varphi_{2}\| _{L^{2-\varepsilon -\lambda}(\Omega)}^{2-\varepsilon}\,d\lambda \biggr]^{\frac {2}{2-\varepsilon }}\,d\varepsilon \\ &\quad \leq C_{4}(\beta,\gamma,K) \int_{0}^{\varepsilon_{0}}\varepsilon ^{\gamma-1} \biggl[ \int_{0}^{\lambda_{0}}\bigl(1+\log|\log\lambda|\bigr)^{-\beta -1}\bigl(\lambda|\log \lambda|\bigr)^{-1}\\ &\qquad{}\times\| \underline{\chi}_{1}-\underline{ \chi}_{2}\| _{L^{2-\varepsilon-\lambda}(\Omega)}^{2-\varepsilon}\,d\lambda \biggr] ^{\frac{2}{2-\varepsilon}}\,d\varepsilon\\ &\quad\leq C_{5}(\beta,\gamma,K) \int_{0}^{\varepsilon_{0}}\varepsilon ^{\gamma-1} \| \underline{\chi}_{1}-\underline{\chi}_{2}\|_{L^{2- \varepsilon} ( \log \log\log L )^{-\beta}(\Omega)}^{2}\,d\varepsilon. \end{aligned}$$

Using again Theorem 3.1 in the last term, we have

$$\begin{aligned} &\|\nabla\varphi_{1}-\nabla \varphi_{2}\|^{2}_{L^{2}(\log L)^{-\gamma}(\log \log \log L)^{-\beta}(\Omega)}\\ &\quad\leq C_{5}( \beta,\gamma,K)|\!|\!|\underline{\chi}_{1}-\underline{\chi }_{2}|\!|\!|^{2}_{L^{2}(\log L)^{-\gamma}(\log\log\log L)^{-\beta}(\Omega )}\\ &\quad\leq C_{6}(\beta,\gamma,K)\|\underline{\chi}_{1}- \underline{\chi }_{2}\| ^{2}_{L^{2}(\log L)^{-\gamma} (\log\log\log L)^{-\beta}(\Omega)} . \end{aligned}$$

 □

Now we are in position to prove the main theorem.

Proof of Theorem 1.1

Since $L^{\widetilde{\Phi}}(\Omega)=L (\log L)^{\delta}(\log\log \log L)^{\frac{\beta}{2}}(\Omega)$ is a subspace of $L(\log L)^{\frac {1}{2}}(\Omega)$ if $\beta\geq0$ and $\delta\geq\frac{1}{2}$, we can ensure (as already observed) that (1.1) has a unique finite energy solution $v\in W^{1,2}_{0}(\Omega)$.

In order to prove Theorem 1.1, we want to apply the regularity result given by Theorem 4.2. To do this, as already showed in the papers [10, 11, 13], and [12], we need to linearize problem (1.1). We will use a linearization procedure introduced in [27] that preserves the ellipticity bounds.

For shortness, we do not give all the details of the linearization procedure, and we refer, for example, to proof of Theorem 1.1 in [11]. So we know that there exists a symmetric, definite positive, and measurable matrix-valued function $B=B(x)$ such that

$$A(x, \nabla v)= B(x) \nabla v. $$

Then, the unique finite energy solution $v\in W^{1,2}_{0}(\Omega)$ of (1.1) with $f\in L^{\widetilde{\Phi}}(\Omega)$ solves also the following linear problem:

$$\begin{aligned} \textstyle\begin{cases} -\operatorname {div}B(x)\nabla v=f & \mbox{in } \Omega, \\ v=0 & \mbox{on } \partial\Omega, \end{cases}\displaystyle \end{aligned}$$

(4.6)

that is,

$$ \int_{\Omega}B(x)\nabla v\nabla\varphi= \int_{\Omega}f\varphi, \quad\forall\varphi\in W^{1,2}_{0}( \Omega) . $$

(4.7)

The case $\boldsymbol{\beta= 0}$ and $\boldsymbol{\frac{1}{2}\leq \delta\leq1}$ has been proved in [10].

The case $\boldsymbol{\beta> 0}$ and $\boldsymbol{\delta=\frac {1}{2}}$ has been proved in [13].

Now, if $\boldsymbol{\beta\geq0}$ and $\boldsymbol{\delta>\frac {1}{2}}$, then we fix $\underline{\chi}\in C^{1}(\overline{\Omega})$ such that

$$\|\underline{\chi}\|_{L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta }(\Omega; \mathbb {R}^{2})}\leq1, $$

and we consider the unique finite energy solution φ to the linear Dirichlet problem

$$\begin{aligned} \textstyle\begin{cases} -\operatorname {div}B(x)\nabla\varphi=\operatorname {div}\underline{\chi} & \mbox{in } \Omega, \\ \varphi=0 & \mbox{on } \partial\Omega, \end{cases}\displaystyle \end{aligned}$$

where $B(x)$ is the matrix given by the linearization procedure. By Theorem 4.2 we have

$$\begin{aligned} &\|\nabla\varphi\|_{L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta }(\Omega)}\\ &\quad\leq C(\beta,\delta,K)\|\underline{\chi}\|_{L^{2}(\log L)^{-(2\delta -1)}(\log\log\log L)^{-\beta}(\Omega)}\leq C(\beta,\delta,K), \end{aligned}$$

and so, using Lemma 2.4, we obtain

$$ \|\varphi\|_{L^{\Phi}(\Omega)}\leq C_{1}(\beta,\delta,K) , $$

(4.8)

where $\Phi(s)\cong e^{s^{\frac{1}{\delta}} ( \log \log s )^{-\frac{\beta}{2\delta}}}$, and $C_{1}(\beta,K)$ is another constant depending only on β, δ, and K.

Thanks to the fact that v satisfies the linear problem (4.6) and that $B(x)$ is a symmetric matrix, using Lemma 2.3 and the Hölder inequality between the complementary spaces $L^{\Phi}(\Omega)$ and $L^{\widetilde{\Phi}}(\Omega)$, by (4.8) we obtain that, for any $\underline{\chi}\in C^{1}(\overline{\Omega}; \mathbb {R}^{2})$ such that ${\|\underline {\chi }\|_{L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta}(\Omega )}\leq 1}$, we have

$$\begin{aligned} \biggl|\int_{\Omega}\nabla v\cdot\underline{\chi} \biggr|&= \biggl| \int _{\Omega}v \operatorname {div}\underline {\chi} \biggr| \\ &= \biggl| \int_{\Omega}v \operatorname {div}\bigl(B(x)\nabla\varphi\bigr) \biggr|= \biggl| \int_{\Omega}B(x)\nabla v\cdot\nabla\varphi \biggr| \\ &= \biggl|\int_{\Omega}f\varphi \biggr|\leq C_{2}(\beta,\delta)\| \varphi\| _{L^{\Phi }(\Omega)}\|f\|_{L(\log L)^{\delta}(\log\log\log L)^{\frac{\beta }{2}}(\Omega)} \\ &\leq C_{2}(\beta,\delta,K)\|f\|_{L(\log L)^{\delta}(\log\log\log L)^{\frac{\beta}{2}}(\Omega)} , \end{aligned}$$

(4.9)

where $C_{2}(\beta,\delta,K)$ is a constant that depends only on β, δ, and K.

By Theorem 2.1 the dual space of $L^{2}(\log L)^{-(2\delta -1)}(\log \log\log L)^{-\beta}(\Omega)$ is $L^{2}(\log L)^{2\delta-1} (\log\log \log L)^{\beta}(\Omega)$.

Now, since $C^{1}(\overline{\Omega}; \mathbb {R}^{2})$ is dense in $L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta}(\Omega)$ (see [20], Theorem 8.20 and [23], Corollary 5), passing to the supremum in (4.9) under the conditions $\underline{\chi}\in C^{1}(\overline {\Omega}; \mathbb {R}^{2})$, ${\|\underline{\chi}\|_{L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta}(\Omega; \mathbb {R}^{2})}\leq 1}$, we obtain

$$\|\nabla v\|_{L^{2}(\log L)^{2\delta-1}(\log\log\log L)^{\beta }(\Omega )}\leq c(\beta,\delta,K)\|f\|_{L(\log L)^{\delta}(\log\log\log L)^{\frac {\beta}{2}}(\Omega)}, $$

as desired. □

Remark 4.3

In [27], it was proved that the linearization procedure holds in any dimension with the following ellipticity bounds:

$$|\xi|^{2}+ \bigl|A(x, \xi)\bigr|^{2} \leq \biggl( K+ \frac{1}{K} \biggr) \bigl\langle A(x, \xi), \xi\bigr\rangle ,\quad\xi\in \mathbb {R}^{n}, \mbox{ a.e. }x\in\Omega. $$

We would like to point out that the linear growth of $A(x, \xi)$ with respect to ξ is absolutely essential for the previous results. The main difficulty with the n-harmonic-type equations ($n \neq2$) is due to the lack of uniqueness for very weak solutions.