1 Introduction

In this note we consider the following nonlinear Dirichlet problem in divergence form

$$ \left\{ \begin{array}{*{20}l} {\text{div}}\left({\mathcal{A}}(x,\nabla u)+{\mathcal{B}}(x,u)\right) = {\text{div }}{\mathcal{F}} & \quad \text{ in } \Omega \\ u=0 &\quad \text{ on } \partial \Omega\end{array} \right. $$
(1.1)

where \(\Omega \) is a bounded Lipschitz domain of \( {{\mathbb {R}}}^{N}\), \(N>2\) and \({\mathcal {F}}=({\mathcal {F}}_1, {\mathcal {F}}_2,\ldots ,{\mathcal {F}}_N)\) is a vector field defined in \(\Omega \).

In last years there has been a growing interest in studying problem as (1.1) also motivated by the fact that it appears in a very wide range of context. The model case of our problem (1.1) is given by

$$ \left\{ \begin{array}{ll} {\text {div}}\;\left[ \beta (x)\nabla u (x) + \left( {\mathbb {B}} \frac{x}{|x|^2} +h (x) \right) u(x)\right] = {\text {div}}\;{\mathcal {F}}(x), &{}\\ u=0 \quad \text{ on } \partial \Omega .&{} \\ \end{array} \right. $$

where \(0\in \Omega \), \(\beta \) belongs to the John–Nirenberg space of functions with bounded mean oscillation (BMO), \({\mathbb {B}}\) is a positive constant and \(h \in {\mathcal {L}}^{\infty }(\Omega )\). It appears for example in the stationary diffusion–convection problems, and for \({\mathcal {F}}=0\) gives the (stationary) Fokker–Plank equation derived in the statistical description of the Brownian motion of a particle in a fluid, it is closely related to the Black–Scholes equation and other equations in financial mathematics. We refer to [34] for applications to the mean field games theory.

Here we assume that

  • \( {\mathcal {A}}: \Omega \times {{\mathbb {R}}}^{N}\rightarrow {{\mathbb {R}}}^{N}\) is a Carathéodory function i.e.

    $$ x \rightarrow {\mathcal {A}}(x ,\xi ) \text{ is } \text{ measurable } \text{ for } \text{ any } \xi \in {{\mathbb {R}}}^{N}, $$
    (1.2)
    $$ \xi \rightarrow {\mathcal {A}}(x,\xi ) \text{ is } \text{ continuous } \text{ for } \text{ almost } \text{ every } x \in \Omega , $$
    (1.3)

  • there exists a real function \(\beta (x)\geqslant \lambda _0>0\) belonging to the space of functions with bounded mean oscillation, \(\beta \in BMO(\Omega )\), see Sect. 2.3, such that

    $$ | {\mathcal {A}} (x, \xi ) - {\mathcal {A}} (x, \eta ) | \leqslant \Lambda \beta (x) |\xi -\eta | $$
    (1.4)
    $$\begin{aligned}&\beta (x)|\xi -\eta |^2 \leqslant \langle {\mathcal {A}}(x, \xi )- {\mathcal {A}} (x, \eta ), \xi -\eta \rangle \end{aligned}$$
    (1.5)
    $$\begin{aligned}&{\mathcal {A}}(x,0)=0, \end{aligned}$$
    (1.6)

    for almost every \(x \in \Omega \) and for any vectors \(\xi \) and \(\eta \) in \({\mathbb {R}}^N\), where \(\Lambda \geqslant 1\) is a constant,

  • \( {\mathcal {B}}:\Omega \times {\mathbb {R}} \rightarrow {{\mathbb {R}}}^{N}\) is a Carathéodory function verifying the following two properties:

    1. (i)

      there exists a non negative function \(b:\Omega \rightarrow {\mathbb {R}}_+\) in the Weak-\(L^N\) space (or Marcinkiewicz space), and we write \(b\in {\mathcal {L}}^{N,\infty }(\Omega ), \) such that

      $$\begin{aligned} |{\mathcal {B}}(x,s)- {\mathcal {B}}(x,t) | \leqslant b(x)|s-t|, \end{aligned}$$
      (1.7)

      for a.e. \(x\in \Omega \) and for every \(s,t\in {\mathbb {R}}\).

    2. (ii)
      $$\begin{aligned} {\mathcal {B}}(x,0) =0,\qquad \text{ for } \text{ a.e. } x\in \Omega . \end{aligned}$$
      (1.8)

    For the definition of the involved functional spaces we refer the reader to Sect. 2.

Definition 1.1

For every vector field \({\mathcal {F}} \in {\mathcal {L}}^p(\Omega ,{{\mathbb {R}}}^{N})\), \(1\leqslant p\leqslant 2\), we say that \(u\in {\mathcal {W}}_0^{1,p} (\Omega )\) is a solution of the Dirichlet problem (1.1) provided

$$\begin{aligned} \int _\Omega \langle {\mathcal {A}}(x,\nabla u) +{\mathcal {B}}(x,u), \nabla \varphi \rangle \, dx =\int _\Omega \left\langle {\mathcal {F}}, \nabla \varphi \right\rangle \,dx, \qquad \forall \varphi \in C^\infty _0(\Omega ). \end{aligned}$$

Elliptic PDE’s with BMO or VMO assumptions on the coefficients (the space VMO is defined as the closure in BMO of the subspace of uniformly continuous functions) has been considered initially in [12, 15], and more recently in [7, 8, 10, 19, 24, 29, 35, 37].

In particular, for uniformly elliptic equations \( {\text {div}}\;( \beta (x)Du(x)) = {\text {div}}\;{\mathcal {F}}(x)\) with \( \beta (x) \in VMO\), a priori estimates for the gradient of solutions were proved by Di Fazio [15] by using explicit representation formulas involving singular integrals and their commutators. Similar results were obtained by Iwaniec and Sbordone in [29] by a different approach.

On the other hand, as is well known, functions in VMO have a number of nice properties which are not satisfied by general BMO-function.

When \(b=0\), in [10] existence and uniqueness of very weak solutions \(\,\, u\in {\mathcal {W}}^{1,p}_0\) to problem (1.1) for \(p\leqslant 2 \) sufficiently close to 2 have been obtained. In this case, in [38] a detailed consideration of the question about the boundedness of bilinear forms corresponding to second order divergence elliptic problems with BMO coefficients is provided.

Under the assumption \(b\in {\mathcal {L}}^{N,\infty }(\Omega )\) existence and uniqueness of solutions \(u\in {\mathcal {W}}^{1,p}_0(\Omega )\) where p is close to two has been obtained in [35, 37].

It is worth to point out that the only assumption \(b\in \mathcal{L}^{N,\infty }(\Omega )\) does not guarantee the existence of a solution to (1.1) also when \(\beta \) is bounded (see the example contained in [27]). Since \({\mathcal {L}}^\infty (\Omega )\) is not dense in \({\mathcal {L}}^{N,\infty }(\Omega )\), it is meaningful to consider the distance with respect to the \({\mathcal {L}}^{N,\infty }-norm\) of a function \(b \in {\mathcal {L}}^{N,\infty }(\Omega )\) to \(\mathcal L^{\infty }(\Omega )\). As usual, the distance of a given function \(f\in {\mathcal {L}}^{N,\infty }(\Omega )\) to \({\mathcal {L}}^\infty (\Omega )\) is defined as

$$\begin{aligned} dist_{{\mathcal {L}}^{N,\infty } (\Omega ) }(f,{\mathcal {L}}^\infty )=\inf _{g\in {\mathcal {L}}^\infty (\Omega )} \Vert f-g\Vert _{N,\infty }. \end{aligned}$$

The distance to \({\mathcal {L}}^\infty \) in some function spaces is studied in [9].

Our main interest in this note is to show how the regularity of \({\mathcal {F}}\) reflected to the gradient of the weak solution in \({\mathcal {L}}^2 \log ^{-\alpha } {\mathcal {L}}(\Omega )\), \(\alpha >0\).

Dirichlet problems assuming that \({\mathcal {F}} \) belongs to Orlicz–Zygmund space \({\mathcal {L}}^plog^{-\alpha }{\mathcal {L}}(\Omega , {\mathbb {R}}^N) \supset {\mathcal {L}}^p \), \(\alpha >0\) has been studied in [20].

We prove the following main result:

Theorem 1.2

Let assumptions (1.2)–(1.8) be verified. There exists a positive constant \(d=d( \lambda _0, \Lambda , \Vert \beta \Vert _{BMO}, N)\) such that if

$$\begin{aligned} dist_{{\mathcal {L}}^{N,\infty }(\Omega )} (b, {\mathcal {L}}^\infty )<d \end{aligned}$$

then, for every \(\alpha >0\) and every \({\mathcal {F}}\in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )\), problem (1.1) admits a unique weak solution \(u\in W^{1,1}_0 (\Omega ) \) such that \(\nabla u \in {\mathcal {L}}^2 \log ^{-\alpha } {\mathcal {L}}(\Omega )\). There exists a constant \(C>0 \) depending on \(\alpha , \lambda _0,\Lambda ,N, \,\Omega , \Vert \beta \Vert _{BMO}\) such that

$$\begin{aligned} \Vert \nabla u\Vert _{{\mathcal {L}}^2{log}^{-\alpha } {\mathcal {L}}(\Omega ,{{\mathbb {R}}}^{N})} \leqslant C\left( \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ,{\mathbb {R}}^N )} + e^{ C \Vert \mathcal F \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ,{\mathbb {R}}^N )} } \right) . \end{aligned}$$
(1.9)

Note that here we are concerned with the case of dimension \(N>2\), since in the plane the theory of elliptic equations is in many respects simpler and more general.

When \(\beta \in {\mathcal {L}}^{\infty }(\Omega )\) assumption (1.2) has been considered in [22] for the linear case \({\mathcal {A}} (x,\nabla u) ={\mathcal {A}}(x)\nabla u\), and in [25,26,27] for the nonlinear case. We also recover problems considered in [4, 31, 36].

Regularity results for solutions to Dirichlet problems for second order elliptic equations with unbounded coefficients have been also obtained in [11, 14, 16,17,18].

To spend some words on the proof, one of the main tool to obtain a priori estimate (1.9) is a refined version of a continuity estimate due to [5, 10, 21]. More precisely, we obtain estimates depending on the BMO norm of the growth coefficient \(\beta \) and not on its \(L^\infty \)-norm. Then we use a suitable approximation argument.

Our result applies when b belongs to the Lorentz space \( {\mathcal {L}}^{N,q}(\Omega )\) with \(1\leqslant q<+\infty \), without any smallness assumptions on the norm of b. Indeed, since \(L^{\infty }(\Omega )\) is dense in \( {\mathcal {L}}^{N,q}(\Omega )\) whenever \(1\leqslant q<+\infty \), in this case we have \(dist_{\mathcal L^{N,\infty }(\Omega )} (b, {\mathcal {L}}^\infty )=0\). Precisely,

Corollary 1.3

Assume that (1.2)–(1.8) holds and let

$$\begin{aligned} b\in {\mathcal {L}}^{N,q}(\Omega ),\quad \text{ for } \text{ some } 1\leqslant q<+\infty . \end{aligned}$$

Then, for every \(\alpha >0\) and every \({\mathcal {F}}\in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )\), problem (1.1) admits a unique weak solution \(u\in W^{1,1}_0 (\Omega ) \) such that \(\nabla u \in {\mathcal {L}}^2 \log ^{-\alpha } {\mathcal {L}}(\Omega )\). Moreover estimate (1.9) holds true.

More in general,

Corollary 1.4

Assume that (1.2)–(1.8) holds and let

$$\begin{aligned} b\in {\mathcal {X}}(\Omega ) \end{aligned}$$

where \(L^\infty (\Omega )\subseteq {\mathcal {X}} (\Omega ) \subset \mathcal L^{N,\infty } (\Omega )\) is any Banach vector space continuously imbedded in \({\mathcal {L}}^{N,\infty }(\Omega )\) and where \({\mathcal {L}}^{\infty }(\Omega )\) is dense. Then, for every \(\alpha >0\) and every \({\mathcal {F}}\in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )\), problem (1.1) admits a unique weak solution \(u\in W^{1,1}_0 (\Omega ) \) such that \(\nabla u \in {\mathcal {L}}^2 \log ^{-\alpha } \mathcal L(\Omega )\). Moreover estimate (1.9) holds true.

The paper is organized as follows. Section 2 will be devoted to the functional spaces. In particular, in Sect. 2.1 we shall give the definition and properties of Lorenz spaces, in Sect. 2.2 we shall introduce the Zygmund spaces and in Sect. 2.3 we shall recall the BMO spaces. In Sect. 3 we shall state and prove a continuity estimate and regularity result for solutions of (1.1) when the norm of \(b\in {\mathcal {L}}^{N,\infty }(\Omega )\) is sufficiently small. Finally, the proof of Theorem 1.2 is contained in Sect. 4.

2 Some functional space

In this section we recall the definitions of some known function spaces. We refer the reader to [2, 13, 33] for more details.

From now on we denote by \(\Omega \subset {{\mathbb {R}}}^{N}\), \(N>2\), a bounded Lipschitz domain. Moreover, for any measurable function f continuamente defined in \(\Omega \) and \(1\le p<\infty ,\) we set

where the barred integral denotes the average, that is .

2.1 Lorentz spaces

Assume \(1<p,q<+\infty \) and let \(\Omega \subset {{\mathbb {R}}}^{N}\) be a bounded Lipschitz domain. The Lorentz space \({\mathcal {L}}^{p,q}(\Omega )\) is the set of all measurable functions f defined on \(\Omega \) for which

$$\begin{aligned} \Vert f\Vert _{p,q}^q=p\int _{0}^{+\infty } |\Omega _t|^{\frac{q}{p}}t^{q-1}\,dt<+\infty , \end{aligned}$$

where \(\Omega _t= \left\{ x\in \Omega : |f(x)|>t \right\} \).

Note that \({\mathcal {L}}^{p,q}(\Omega )\) is a Banach space with respect to the norm \(||\cdot ||_{p,q}\) (see [32]).

For \(p=q\), the Lorentz space \({\mathcal {L}}^{p,p}(\Omega )\) reduces to the standard Lebesgue space \({\mathcal {L}}^p(\Omega )\). For \(q=+\infty \), the class \({\mathcal {L}}^{p,\infty }(\Omega )\) consists of all measurable functions f defined on \(\Omega \) such that

$$\begin{aligned} ||f||^p_{p,\infty }=\sup _{t>0}t^{ p}|\Omega _t|<+\infty \end{aligned}$$

and it coincides with the Marcinkiewicz class, weak-\({\mathcal {L}}^p\).

We observe that \({\mathcal {L}}^{p,\infty }(\Omega ) \supset {\mathcal {L}}^p(\Omega )\). For example, if \(\Omega \subset {\mathbb {R}}^N\) contains the origin, the function \(f(x)=\vert x \vert ^{-\frac{N}{p}}\not \in {\mathcal {L}}^p(\Omega )\) but \(f\in \mathcal L^{p, \infty }(\Omega )\) with \(\vert \vert f \vert \vert _{p,\infty }^p=\omega _N\), where \(\omega _N\) stands for the Lebesgue measure of the unit ball of \({\mathbb {R}}^N.\)

We just recall some properties of this spaces:

  • for \(1<q<p<r\leqslant +\infty ,\) the following inclusions hold:

    $$\begin{aligned} {\mathcal {L}}^r (\Omega ) \subset {\mathcal {L}}^{p,q}(\Omega )\subset \mathcal L^{p,p}(\Omega )\equiv {\mathcal {L}}^p(\Omega )\subset {\mathcal {L}}^{p,r} (\Omega ) \subset {\mathcal {L}}^{p,\infty }(\Omega )\subset {\mathcal {L}}^q(\Omega ); \end{aligned}$$

  • given \(1<p,s<+\infty \) and \(1<q\leqslant +\infty \) it holds

    $$\begin{aligned} \Vert \,|f|^s\Vert _{p,q}=\Vert f\Vert _{s p,s q}^{s }; \end{aligned}$$

  • Hölder inequality in Lorentz spaces: assume \(1<p<+\infty \), \(1\leqslant q\leqslant +\infty \) and \(\frac{1}{p}+\frac{1}{p'}=1\), \(\frac{1}{q}+\frac{1}{q'}=1\) (for \(q=+\infty \) we assume \(q'=1\)). If \(f\in {\mathcal {L}}^{p,q}(\Omega )\) and \(g\in {\mathcal {L}}^{p',q'}(\Omega )\), then

    $$\begin{aligned} \int _{\Omega }|f(x)g(x)|\text {d} x\leqslant ||f||_{p,q}||g||_{p',q'}. \end{aligned}$$

The following Sobolev embedding theorem in Lorentz spaces holds (see [1, 32]):

Theorem 2.1

Let \(1<p<N\) and \(1\leqslant q\leqslant p\). Then, every function \(f\in {\mathcal {W}}_0^{1,1}(\Omega )\) verifying \(|\nabla f|\in \mathcal L^{p,q}(\Omega )\) belongs to \({\mathcal {L}}^{p^*,q}(\Omega )\), where \(p^*=\frac{Np}{N-p}\), and

$$\begin{aligned} ||f||_{p^*,q}\leqslant S_p||\nabla f||_{p,q}. \end{aligned}$$
(2.1)

Here \(S_p=C(N)\frac{p}{N-p}\).

We remark that \({\mathcal {L}}^\infty (\Omega )\) is not dense in \(\mathcal L^{p,\infty }(\Omega )\), \(p\in {}]1, +\infty [\). We define the distance of a given function \(f\in {\mathcal {L}}^{p,\infty }(\Omega )\) to \(\mathcal L^\infty (\Omega )\) as

$$\begin{aligned} dist_{{\mathcal {L}}^{p,\infty } (\Omega ) }(f,{\mathcal {L}}^\infty )=\inf _{g\in {\mathcal {L}}^\infty (\Omega )} \Vert f-g\Vert _{p,\infty }. \end{aligned}$$

Note that, since \(\Vert ~\Vert _{p,\infty }\) is not a norm, \(dist_{\mathcal L^{p,\infty } (\Omega ) }\) is just equivalent to a metric.

For \(M>0\), we consider the truncation operator

$$\begin{aligned} T_M(y)=\frac{y}{|y|}\min \{|y|, M\} \end{aligned}$$
(2.2)

and by [9] one can see that

$$\begin{aligned} dist_{{\mathcal {L}}^{p,\infty }(\Omega ) }(f,{\mathcal {L}}^\infty )=\lim _{M \rightarrow +\infty }\Vert f-T_M f\Vert _{p,\infty }. \end{aligned}$$
(2.3)

Here and below, for a fixed value \(M>0\) we set

$$\begin{aligned} \theta _{f,M}(x)= \left\{ \begin{array}{ll} \frac{T_M f(x)}{f(x)} &{}\quad \hbox { if } f(x)\ne 0 \\ 1 &{}\quad \hbox { if }f(x)=0 . \\ \end{array} \right. \end{aligned}$$
(2.4)

2.2 Zygmund spaces

We recall some useful definitions and properties of Zygmund spaces. We refer to [2, 13, 20, 33], for more details.

The Zygmund space \( {\mathcal {L}}^p {log}^{\beta } \mathcal L(\Omega )\), for \(1<p<+\infty \) and \(\beta \in {\mathbb {R}}\), is the Orlicz space generated by the function

$$\begin{aligned} \phi (t)=t^p\log ^{\beta }(a+t), \quad t\ge 0, \end{aligned}$$

where \(a\ge e\) is a suitably large constant, so that \(\phi \) is increasing and convex on \([0, +\infty [\). Precisely we denote by \( {\mathcal {L}}^p {log}^{\beta } {\mathcal {L}}(\Omega )\) the set of all measurable functions f on \(\Omega \) such that

$$\begin{aligned} \int _{\Omega } \vert f\vert ^p log^{\beta } (a+\vert f \vert ) dx <+ \infty . \end{aligned}$$

\({\mathcal {L}}^p {log}^{\beta } {\mathcal {L}}(\Omega )\) is a Banach space equipped with the following Luxemburg norm

$$\begin{aligned}{}[f]_{ {\mathcal {L}}^p {log}^{\beta } {\mathcal {L}}(\Omega )}=inf \left\{ \lambda >0 : \int _{\Omega } \phi \left( \frac{\vert f\vert }{\lambda } \right) dx \le 1 \right\} . \end{aligned}$$
(2.5)

We observe that if \(\vert \Omega \vert \) is finite and \(p>q\), then the following inclusion holds

$$\begin{aligned} {\mathcal {L}}^p log^{\beta } {\mathcal {L}}(\Omega ) \subset {\mathcal {L}}^q log^{\beta '} {\mathcal {L}}(\Omega ) \quad \quad \forall \beta , \beta ' \in {\mathbb {R}}. \end{aligned}$$

For \(\alpha >0\) we also consider the following quantities

$$\begin{aligned} \vert \vert f \vert \vert _{ {\mathcal {L}}^p {log}^{-\alpha } \mathcal L(\Omega )}= \left\{ \int _0^{\varepsilon _0} \varepsilon ^{\alpha -1} \vert \vert f \vert \vert ^p_{p-\varepsilon } d\varepsilon \right\} ^{\frac{1}{p}} \end{aligned}$$
(2.6)

where \(0<\varepsilon \le p-1\) is fixed.

The following result establish the equivalence between the norms (2.5) and (2.6) respectively (see [20]).

Proposition 2.2

The measurable function \(f:\Omega \rightarrow {\mathbb {R}}\) belongs to \( \mathcal L^p {log}^{-\alpha } {\mathcal {L}}(\Omega )\) if and only if

$$\begin{aligned} \vert \vert f \vert \vert _{ {\mathcal {L}}^p {log}^{-\alpha } \mathcal L(\Omega )}<+\infty . \end{aligned}$$

Moreover, \(\vert \vert f \vert \vert _{ {\mathcal {L}}^p {log}^{-\alpha } {\mathcal {L}}(\Omega )}\) is a norm equivalent to the Luxemburg one, that is, there exist constants \(C_i=C_i(q, \alpha , a, \varepsilon _0)\), \(i=1,2\), such that for all \(f\in {\mathcal {L}}^p {log}^{-\alpha } {\mathcal {L}}(\Omega )\)

$$\begin{aligned} C_1 [f]_{ {\mathcal {L}}^p {log}^{-\alpha } {\mathcal {L}}(\Omega )} \le \vert \vert f \vert \vert _{ {\mathcal {L}}^p {log}^{-\alpha } {\mathcal {L}}(\Omega )} \le C_2[f]_{ {\mathcal {L}}^p {log}^{-\alpha } {\mathcal {L}}(\Omega )}. \end{aligned}$$

For \(p>1\), \( \alpha >0\) and \(0<\varepsilon \leqslant p-1\) the following continuous inclusions hold true:

$$\begin{aligned} {\mathcal {L}}^p {log}^{\alpha } {\mathcal {L}}(\Omega )\subset \mathcal L^p(\Omega ) \subset {\mathcal {L}}^p {log}^{-\alpha } {\mathcal {L}}(\Omega ) \subset {\mathcal {L}}^{p-\varepsilon }(\Omega ). \end{aligned}$$
(2.7)

The Zygmund- Sobolev space is defined, for \(1<p<+\infty \) and \(\beta \in {\mathbb {R}},\) as

$$\begin{aligned} {\mathcal {W}}^{1,p}_0 {log}^{\beta } {\mathcal {L}}(\Omega )=\{ f\in {\mathcal {W}}_0^{1,1}(\Omega ) \; \vert \; \vert \vert \nabla f\vert \vert _{ {\mathcal {L}}^p {log}^{\beta } {\mathcal {L}}(\Omega ) }<+\infty \}. \end{aligned}$$

We conclude this section by recalling the following Sobolev embedding theorem in Zygmund spaces (see [13])

Theorem 2.3

Let \(1<p<N\) and \(q\in {\mathbb {R}}\). Then, every function \(f\in {\mathcal {W}}_0^{1,1}(\Omega )\) verifying \(|\nabla f|\in \mathcal L^{p}log^q{\mathcal {L}} (\Omega )\) actually belongs to \(\mathcal L^{p^*}\log ^r {\mathcal {L}}(\Omega )\), where \(p^*=\frac{Np}{N-p}\) and \(r=\frac{Nq}{N-p}\). Moreover

$$\begin{aligned} ||f||_{ {\mathcal {L}}^{p^*} {log}^{r} {\mathcal {L}}(\Omega )}\leqslant S||\nabla f||_{{\mathcal {L}}^{p} {log}^{q} {\mathcal {L}}(\Omega )} \end{aligned}$$
(2.8)

where S is a positive constant depending only on \(N, \Omega , p \) and q.

2.3 Bounded mean oscillation spaces

We say that \(\beta \in {\mathcal {L}}^1(\Omega )\) is a function with bounded mean oscillation on \(\Omega \), and we write \(\beta \in BMO(\Omega )\), if

where the supremum is taken over all cubes \(Q\subset \Omega \) with sides which are parallel to the coordinate axis and

We recall the following Lemma (see [30])

Lemma 2.4

Let \(\Omega \) be a bounded domain of \({{\mathbb {R}}}^{n}\) and let \(\beta \geqslant 0\) be a function in \(BMO(\Omega )\). Then, for every \(1<p<+\infty \),

$$\begin{aligned} \int _\Omega |\beta (x) |^p \text {dx} \leqslant p |\Omega |^{\frac{1}{p'}} C(\beta ,\Omega ). \end{aligned}$$

3 Preliminary results

This section contains some preliminary result that will be useful in the sequel. From now on we shall assume \(\alpha >0.\)

We start by proving the following continuity estimate (see [10, 20, 21] for the case \(b=0\)).

Proposition 3.1

Assume (1.2)–(1.8) and let \(\beta \in \mathcal {L}^\infty (\Omega )\). There exist two positive constants \(d=d(N,\Lambda ,\Vert \beta \Vert _{BMO},\lambda _0)\) and \(C=C(\alpha , N,\Lambda ,\Vert \beta \Vert _{BMO},\lambda _0)\), with d decreasing and C increasing with respect to the BMO-norm of \(\beta \), such that if

$$\begin{aligned} \Vert b\Vert _{N,\infty }<d \end{aligned}$$

then, for every \({\mathcal {F}},{\mathcal {G}}\in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )\) and \(u,v \in {\mathcal {W}}_0^{1,2}log^{-\alpha }{\mathcal {L}}(\Omega )\) solutions to problems

$$\begin{aligned} \left\{ \begin{array}{ll} {\text {div}}\;[ {\mathcal {A}}(x,\nabla u)+{\mathcal {B}} (x,u)]= {\text {div}}\;{\mathcal {F}} &{}\quad \hbox { in } \Omega \\ u=0 &{}\quad \hbox { on } \partial \Omega \\ \end{array} \right. \end{aligned}$$
(3.1)

and

$$\begin{aligned} \left\{ \begin{array}{ll} {\text {div}}\;[ {\mathcal {A}}(x,\nabla v)+{\mathcal {B}}(x,v)]= {\text {div}}\;{\mathcal {G}} &{}\quad \hbox { in } \Omega \\ v=0 &{}\quad \hbox { on } \Omega \\ \end{array} \right. \end{aligned}$$
(3.2)

respectively, the following estimate holds

$$\begin{aligned} \Vert \nabla u-\nabla v \Vert _{ {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega , {\mathbb {R}}^N )} \leqslant C \Vert {\mathcal {F}}-{\mathcal {G}} \Vert _{ {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )}. \end{aligned}$$
(3.3)

Proof

Let \( {\mathcal {F}}, {\mathcal {G}} \in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )\) and let \(u,v \in \mathcal W_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\) be the corresponding solutions to problems (3.1) and (3.2). We observe that by (2.6) \({\mathcal {F}}, {\mathcal {G}}\in \mathcal L^{2-\varepsilon }(\Omega , {\mathbb {R}}^N )\), for every \(\varepsilon \in (0,1].\) Hence, using Proposition 4.1 of [37] there exist three positive constants \(\varepsilon _0\), d,  and C, depending only on \(N,\Lambda ,\Vert \beta \Vert _{BMO} \) and \(\lambda _0\), with \(\varepsilon _0\leqslant \frac{4}{N+2}\), \(\varepsilon _0 \) and d decreasing and C increasing with respect to the BMO-norm of \(\beta \), such that if

$$\begin{aligned} \Vert b\Vert _{N,\infty }<d \end{aligned}$$

then, for every \( 0<\varepsilon <\varepsilon _0,\)

$$\begin{aligned} \Vert \nabla u-\nabla v \Vert _{ {\mathcal {L}}^ {2-\varepsilon }(\Omega , \mathbb {R}^N )} \le C \Vert {\mathcal {F}}-{\mathcal {G}} \Vert _{ \mathcal L^{2-\varepsilon }(\Omega , {\mathbb {R}}^N )} . \end{aligned}$$
(3.4)

Here C is independent on \(\varepsilon \). Then, multiplying by \(\varepsilon ^{\alpha -1}\) both sides of (3.4) and integrating with respect to \(\varepsilon \) on the interval \((0,\varepsilon _0)\), we obtain

$$\begin{aligned} \int _0^{\varepsilon _0} \varepsilon ^{\alpha -1} \Vert \nabla u-\nabla v \Vert _{ L^ {2-\varepsilon }(\Omega , {\mathbb {R}}^N )} \;d\varepsilon \le C \int _0^{\varepsilon _0} \varepsilon ^{\alpha -1} \Vert {\mathcal {F}}-{\mathcal {G}} \Vert _{ L^{2-\varepsilon }(\Omega , {\mathbb {R}}^N )} \; d\varepsilon . \end{aligned}$$
(3.5)

By Propositions 2.2 and (3.5) we get (3.3). \(\square \)

As a consequence of the previous result, we have the following

Corollary 3.2

Assume (1.2)–(1.8) and \(\beta \in \mathcal L^\infty (\Omega )\). Let \(d\!=\!d(\lambda _0, N, \Lambda , \Vert \beta \Vert _{BMO})\) be the constant obtained in Proposition3.1. If

$$\begin{aligned} \Vert b\Vert _{N,\infty }<d \end{aligned}$$

then, for every \({\mathcal {F}} \in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N)\), problem (1.1) admits a solution \(u\in {\mathcal {W}}_0^{1,2} log^{-\alpha }{\mathcal {L}}(\Omega )\) and such solution is unique.

Proof

Since the uniqueness of solutions \(u\in \mathcal W_0^{1,2}log^{-\alpha }{\mathcal {L}}(\Omega )\) to problem (1.1) is an obvious consequence of (3.3), we just prove the existence.

To this aim, let \({\mathcal {F}} \in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N)\) and \(F_j: \Omega \rightarrow {\mathbb {R}}^N\) be a sequence of functions such that

$$\begin{aligned} F_j\in {\mathcal {L}}^2(\Omega , {{\mathbb {R}}}^{N}) \quad \hbox {and} \quad {\mathcal {F}}_j \rightarrow {\mathcal {F}} \; \hbox {in} \; {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N) \end{aligned}$$

and let \(u_j\in {\mathcal {W}}_0^{1,2}(\Omega )\) be the corresponding solution sequence to (3.1) (for the existence of such solutions see [27, 37]).

By Proposition 3.1 we have

$$\begin{aligned} \Vert \nabla u_j-\nabla u_k \Vert _{ {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega , {\mathbb {R}}^N )} \leqslant C \Vert {\mathcal {F}}_j-{\mathcal {F}}_k \Vert _{ {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )}. \end{aligned}$$
(3.6)

Then \(\{u_j\}_{j\in N}\) is a Cauchy sequence in \( \mathcal W_0^{1,2}log^{-\alpha }{\mathcal {L}}(\Omega )\).

Let \(u\in {\mathcal {W}}_0^{1,2}log^{-\alpha }\mathcal L(\Omega )\) be the limit of \(u_j\). Passing to the limit in (3.6) as \(k\rightarrow +\infty \) we obtain

$$\begin{aligned} \Vert \nabla u_j-\nabla u \Vert _{ {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega , {\mathbb {R}}^N )}\le C \Vert {\mathcal {F}}_j-{\mathcal {F}} \Vert _{ {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )}. \end{aligned}$$
(3.7)

Then, by the equality

$$\begin{aligned} \int _\Omega \langle {\mathcal {A}}(x,\nabla u_j) +{\mathcal {B}}(x,u_j), \nabla \varphi \rangle \, dx =\int _\Omega \left\langle {\mathcal {F}}_j, \nabla \varphi \right\rangle \,dx \quad \forall \varphi \in C_0^{\infty } \end{aligned}$$

letting \(j\rightarrow +\infty \) and using (3.7), we obtain that u is a solution to problem (1.1). \(\square \)

Now, we prove the following higher integrability for the difference of solutions to our problems. The proof relies on ideas contained in [27].

Proposition 3.3

Assume (1.2)–(1.8) and let \(\beta \in \mathcal L^\infty (\Omega )\). Moreover, let \(d\!=\!d(\lambda _0, N, \Lambda , \Vert \beta \Vert _{BMO})\) be the constant obtained in Proposition 3.1. If

$$\begin{aligned} \Vert b\Vert _{N,\infty }<d \end{aligned}$$

then, for every \( {\mathcal {F}}, {\mathcal {G}} \in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )\), denoting by \(u,v \in {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\) the corresponding solutions to problem (3.1) and (3.2) respectively, we have

$$\begin{aligned} ({\mathcal {F}}-{\mathcal {G}})\in {\mathcal {L}}^2(\Omega , {\mathbb {R}}^N) \Longrightarrow (\nabla u-\nabla v) \in {\mathcal {L}}^2(\Omega , {\mathbb {R}}^N). \end{aligned}$$

Proof

Let \({\mathcal {F}}, {\mathcal {G}}\in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )\) and let \( ({\mathcal {F}}-\mathcal G)\in {\mathcal {L}}^2(\Omega ;{\mathbb {R}}^N)\). We consider the extension of \({\mathcal {F}}, {\mathcal {G}}\) to \({\mathbb {R}}^N\) assuming them zero outside \(\Omega .\)

Let \(\{{\mathcal {F}}_n\}_{n\in {\mathbb {N}}}\) and \(\{\mathcal G_n\}_{n\in {\mathbb {N}}}\) be smooth functions satisfying

$$\begin{aligned} {\mathcal {F}}_n\rightarrow {\mathcal {F}} \,\,\, \text{ in } \mathcal L^{2}log^{-\alpha }{\mathcal {L}}(\Omega , {\mathbb {R}}^N) \qquad \,\qquad \,\,\,\,\,\, {\mathcal {G}}_n\rightarrow {\mathcal {G}}\, \,\,\, \text{ in } {\mathcal {L}}^{2}log^{-\alpha }{\mathcal {L}}(\Omega , \mathbb R^N)\,\, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {F}}_n-{\mathcal {G}}_n\rightarrow {\mathcal {F}}-{\mathcal {G}}\,\,\, \text{ in } {\mathcal {L}}^{2}(\Omega , {\mathbb {R}}^N) . \end{aligned}$$

Denote by \(u_n\) and \(v_n\) in \({\mathcal {W}}^{1,2}_0(\Omega )\) the solutions to problem (3.1) and (3.2) corresponding to \({\mathcal {F}}_n\) and \({\mathcal {G}}_n\) respectively.

Using Proposition 4.1 of [37] where we set \(q=2\), it holds

$$\begin{aligned} \Vert \nabla u_n-\nabla v_n\Vert _{{\mathcal {L}}^2(\Omega ,{\mathbb {R}}^N )}\le c\Vert {\mathcal {F}}_n-{\mathcal {G}}_n\Vert _{{\mathcal {L}}^2(\Omega ,{\mathbb {R}}^N)}. \end{aligned}$$

Hence, the sequence \(\{u_n-v_n\}_{n\in N} \) is bounded in \(\mathcal W^{1,2}_0(\Omega ).\) Then, unless to pass to a subsequence, there exists \(w\in {\mathcal {W}}^{1,2}_0(\Omega )\) such that

$$\begin{aligned} u_n-v_n \rightarrow w \text{ weakly } \text{ in } \mathcal W^{1,2}(\Omega ). \end{aligned}$$

On the other hand, by (2.7) and (3.3),

$$\begin{aligned} \Vert \nabla u_n-\nabla u\Vert _{{\mathcal {L}}^2log^{-\alpha }{\mathcal {L}}(\Omega )} \le c\Vert {\mathcal {F}}_n-{\mathcal {F}}\Vert _{{\mathcal {L}}^2log^{-\alpha }{\mathcal {L}}(\Omega )} \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla v_n-\nabla v\Vert _{{\mathcal {L}}^2log^{-\alpha }\mathcal L(\Omega )}\le c\Vert {\mathcal {G}}_n-{\mathcal {G}}\Vert _{{\mathcal {L}}^2log^{-\alpha }{\mathcal {L}}(\Omega )} \end{aligned}$$

so that

$$\begin{aligned} (u_n-v_n )\rightarrow (u-v ) \text{ strongly } \text{ in } {\mathcal {W}}^{1,2}_0 log^{-\alpha }{\mathcal {L}} (\Omega ). \end{aligned}$$

This means

$$\begin{aligned} (u-v)\equiv w\in {\mathcal {W}}^{1,2}_0(\Omega ) \end{aligned}$$

and the proof is completed.

\(\square \)

Remark 3.4

We explicitly note that the continuity estimate and the regularity result stated in Propositions 3.1 and 3.3 obtained for \(\beta \in {\mathcal {L}}^\infty (\Omega )\) can be proved also assuming \(\beta \in BMO\). Indeed, we can use estimate (3.4) obtained in [10] for \(\beta \in BMO\) and similar arguments as before.

4 Proof of Theorem 1.2

In this section we prove our main result Theorem 1.2. We divide our proof into two steps. In the first step we obtain existence of solution to problem (1.1) under the additional assumption \(\beta \in {\mathcal {L}}^\infty (\Omega )\). Here a crucial result is an estimate of the gradient of solutions (see (4.3)) in terms of quantities depending on the BMO-norm of \(\beta \) and not on its \({\mathcal {L}}^{\infty }\)-norm. Then, we remove the additional assumption \(\beta \in {\mathcal {L}}^\infty (\Omega )\) by using an approximation argument.

Before to start, it is worth to recall the following Leray–Schauder fixed point theorem for a continuous and compact mapping of a Banach space into itself (see [23]), since it will be a key tool in our proof.

A continuous mapping between two Banach spaces is called compact if the images of bounded sets are pre-compact.

Theorem 4.1

(Leray–Schauder fixed point theorem) Let \({\mathbb {F}}\) be a continuous and compact mapping of a Banach space X into itself and suppose there exists a constant \(\mathbb {K}\) such that \(\Vert x\Vert _{X} <{\mathbb {K}}\) for all \(x\in X\) and \(t\in [0,1]\) satisfying \(x=t{\mathbb {F}}(x)\). Then \({\mathbb {F}}\) has a fixed point.

Proof of Theorem 1.2

Let \(d=d(\Lambda , \Vert \beta \Vert _{BMO}, \lambda _0,N) \) be the positive constant obtained in Proposition 3.1, and assume that

$$\begin{aligned} dist_{{\mathcal {L}}^{N,\infty } (\Omega ) }(b,{\mathcal {L}}^{\infty }) <d. \end{aligned}$$
(4.1)

By (4.1) and (2.3), we can fix a constant \(M=M (\Lambda ,\Vert \beta \Vert _{BMO}, N,\lambda _0)\geqslant 1\) such that the truncated function \(T_{ M } b\) defined in (2.2) at level M satisfies

$$\begin{aligned} \Vert b-T_{M}b\Vert _{N,\infty }<d. \end{aligned}$$
(4.2)

We rewrite the equation in (1.1) in the following way

$$\begin{aligned} {\text {div}}\;({\mathcal {A}}(x,\nabla u)+(1-\theta (x)){\mathcal {B}}(x,u)) = {\text {div}}\;( {\mathcal {F}} -\theta (x){\mathcal {B}}(x,u)) \end{aligned}$$

where \(\theta = \theta _{b,M}\) is defined in (2.4).

Step 1 Let us firstly assume that \(\beta \in {\mathcal {L}}^{\infty }(\Omega )\). Under this additional assumption we shall prove that for every \( {\mathcal {F}} \in {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )\), there exists a solution \( u\in {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\) to problem (1.1) and that the following a priori estimate holds

$$\begin{aligned} \Vert \nabla u\Vert _{ {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega )}\leqslant {\mathbb {K}} \end{aligned}$$
(4.3)

where \({\mathbb {K}}={\mathbb {K}}(\alpha ,\Lambda ,N, \Vert \beta \Vert _{BMO},\Omega , \Vert {\mathcal {F}}\Vert _{ {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega , {\mathbb {R}}^N )}\), \(\lambda _0)\) is a positive constant increasing with respect to \(\Vert \beta \Vert _{BMO}.\)

So, let \({\mathcal {F}} \in {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega , {\mathbb {R}}^N )\). For every \({\bar{u}}\in {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\) we consider the following problem

$$\begin{aligned} \left\{ \begin{array}{ll} {\text {div}}\;[ {\mathcal {A}}\left( x,{\nabla u}\right) + (1-\theta (x) ) {\mathcal {B}}\left( x, u\right) ] ={\text {div}}\;[ {\mathcal {F}}-\theta (x) {\mathcal {B}} (x,{{\bar{u}}})]&{} \\ u\in {\mathcal {W}}_0^{1,2}log^{-\alpha } {\mathcal {L}}(\Omega ).&{}\\ \end{array} \right. \end{aligned}$$
(4.4)

Observe that by assumptions (1.7), (1.8) and (2.8) we obtain

$$\begin{aligned}{}[{\mathcal {F}}-\theta (\cdot ) {\mathcal {B}} (\cdot ,{{\bar{u}}})] \in \mathcal L^2 {log}^{-\alpha } {\mathcal {L}}(\Omega ). \end{aligned}$$

Moreover, by (4.2) and (2.4)

$$\begin{aligned} \Vert (1-\theta (\cdot ) ) b(\cdot )\Vert _{{\mathcal {L}}^{N,\infty } (\Omega )} = \Vert b(\cdot ) -T_M b(\cdot )\Vert _{{\mathcal {L}}^{N,\infty } (\Omega )} <d. \end{aligned}$$

Hence, by Corollary 3.2, there exists a unique solution \(u\in {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\) to problem (4.4). So, we consider the following operator

$$\begin{aligned} {\mathbb {F}} :{\bar{u}} \in {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega ) \rightarrow u \in {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\, \end{aligned}$$

where \(u:={\mathbb {F}} ({{\bar{u}}}) \) is the unique solution to problem (4.4).

The operator \({\mathbb {F}}\) is a continuous and compact.

To prove the compactness, let \(\{u_j\}_{j\in N}\) be a bounded sequence in \({\mathcal {W}}_0^{1,2} {log}^{-\alpha } \mathcal L(\Omega )\). Then, unless to pass to a subsequence, we have:

$$\begin{aligned}&\bar{u_j}\rightharpoonup {\bar{u}} \; \hbox {in } \; {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\quad \hbox {weakly}\\&\bar{u_j}\rightarrow {\bar{u}} \; \hbox {in } \; {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )\quad \hbox {strongly}. \end{aligned}$$

By Proposition 3.1, (1.7) and (2.4) we obtain

$$\begin{aligned}&\vert \vert \nabla {\mathbb {F}}(\bar{u_j})- \nabla {\mathbb {F}}({\bar{u}}) \vert \vert _{ {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \le C\vert \vert \theta (\cdot ) {\mathcal {B}}(\cdot ,\bar{u_j}) - {\mathcal {B}}(\cdot ,{\bar{u}}) \vert \vert _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \\\\&\quad \le C M \vert \vert \bar{u_j}-{\bar{u}} \vert \vert _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \end{aligned}$$

and this prove the compactness of \({\mathbb {F}}\). The continuity follows by similar arguments.

Now, we shall use the fixed point Theorem 4.1. Precisely, we shall prove that there exists \({\mathbb {K}}>1\) such that \( \forall t\in [0,1]\) and \(\forall u\in {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\), the solution to \(u=t{\mathbb {F}}(u)\) satisfies

$$\begin{aligned} \vert \vert \nabla u\vert \vert _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )}\le {\mathbb {K}}. \end{aligned}$$

Preliminary, we observe that, for \(s\in (0,1]\) sufficiently small, \(s{\mathbb {F}}\) is a contraction on \(\mathcal W_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\).

Indeed, \( \forall {\bar{u}}, {\bar{v}}\in \mathcal W_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\), from Proposition 3.1, (1.7) and (2.8) we have

$$\begin{aligned}&\vert \vert \nabla s {\mathbb {F}}({\bar{u}})- \nabla s\mathbb F({\bar{v}}) \vert \vert _{ {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega )} \le s C \vert \vert {\mathbb {F}}({\bar{u}})- \mathbb F({\bar{v}}) \vert \vert _{ {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega )} \\\\&\quad \le s C \vert \vert \theta (x ) \mathcal B(x , {\bar{u}}) - {\mathcal {B}}(x , {\bar{v}}) \vert \vert _{ {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \le s C M \vert \vert {\bar{u}}-{\bar{v}} \vert \vert _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \\\\&\quad \le s C M \vert \vert {\bar{u}}-{\bar{v}} \vert \vert _{{\mathcal {L}}^{2^*} {log}^{r} {\mathcal {L}}(\Omega )} \le s C M \vert \vert \nabla {\bar{u}}- \nabla {\bar{v}} \vert \vert _{{\mathcal {L}}^{2} {log}^{-\alpha } \mathcal L(\Omega )}. \end{aligned}$$

Hence, for \(s<\frac{1}{CM}\), \(s{\mathbb {F}}(v)\) is a contraction.

Moreover, using again Proposition 3.1, by (1.6) and (1.8) we have

$$\begin{aligned} \left\| \frac{\nabla v}{s}\right\| _{L^2{log}^{-\alpha } \mathcal L(\Omega )}\le & {} C \Vert {\mathcal {F}}-\theta (x)B(x,v)\Vert _{ \mathcal L^2{log}^{-\alpha } {\mathcal {L}}(\Omega )} \\\\\leqslant & {} C\left( \Vert {\mathcal {F}}\Vert _{ {\mathcal {L}}^2{log}^{-\alpha } {\mathcal {L}}(\Omega )} + M \Vert v \Vert _{ {\mathcal {L}}^2{log}^{-\alpha } {\mathcal {L}}(\Omega )}\right) \end{aligned}$$

so that

$$\begin{aligned} \left\| \nabla v\right\| _{ {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega )}\le s C \Vert {\mathcal {F}}\Vert _{ {\mathcal {L}}^2{log}^{-\alpha } {\mathcal {L}}(\Omega )} + s C M \left\| \nabla v\right\| _{ \mathcal L^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \end{aligned}$$

and, for \(s\in (0,1) \) small enough in order to have \( s C M<{1\over 2}\)

$$\begin{aligned} \left\| \nabla v\right\| _{ {\mathcal {L}}^2 {log}^{-\alpha } \mathcal L(\Omega )}\le 2c s \Vert {\mathcal {F}}\Vert _{ {\mathcal {L}}^2{log}^{-\alpha } {\mathcal {L}}(\Omega )}, \end{aligned}$$

with s decreasing with respect to \(\Vert \beta \Vert _{BMO}.\)

This gives

$$\begin{aligned} \left\| \frac{\nabla v}{s}\right\| _{ {\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )}\le C \Vert {\mathcal {F}}\Vert _{ \mathcal L^2{log}^{-\alpha } {\mathcal {L}}(\Omega )} \end{aligned}$$
(4.5)

with C increasing with respect to \(\Vert \beta \Vert _{BMO}\) and independent on s. Hence, we fix a such small value of \(s\in (0,1]\), and we consider the corresponding function \(v=s\mathbb F(v) \in {\mathcal {W}}_0^{1,2} {log}^{-\alpha } {\mathcal {L}}(\Omega )\).

Let \(t\in (0,1]\) and let \(u=t{\mathbb {F}}(u)\). We have

$$\begin{aligned}&\Vert \nabla u\Vert _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \le \left\| \frac{\nabla u}{t} \right\| _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \\\\&\quad \leqslant \left\| \frac{\nabla v}{s}\right\| _{L^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )}+ \,\left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{L^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )}. \end{aligned}$$
(4.6)

Now, we shall estimate the term \( \,\left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )}\).

Using the fact that \(u=t{\mathbb {F}}(u)\) and \(v=s\mathbb F(v)\), by Proposition 3.1, (1.7), (1.8) we have

$$\begin{aligned}&\left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{\mathcal L^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} =\left\| \nabla \mathbb F(u) -\nabla {\mathbb {F}}(v) \right\| _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \\\\&\quad \le \left\| \theta (x ) \left( {\mathcal {B}}(x , u) - {\mathcal {B}}(x , v) \right) \right\| _{{\mathcal {L}}^2 {log}^{-\alpha } {\mathcal {L}}(\Omega )} \\\\&\quad \le C M \vert \vert u-v\vert \vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega )}. \end{aligned}$$
(4.7)

In particular, by (4.7), (2.8) and Proposition 3.3 it follows that

$$\begin{aligned} \left( \frac{\nabla u}{t}-\frac{\nabla v}{s} \right) \in \mathcal L^2(\Omega ). \end{aligned}$$
(4.8)

So, using (2.7), it will be sufficient to estimate the quantity \(\left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{{\mathcal {L}}^2(\Omega )}.\)

Combining (1.4), (1.7) and (4.8) it follows that we can consider test functions \(\varphi \in \mathcal W^{1,2}_0(\Omega )\) in the difference between equations \(u=t\mathbb F(u)\) and \(v=s{\mathbb {F}}(v)\). Precisely, we have that

$$\begin{aligned}&\ \int _\Omega \left\langle {\mathcal {A}} \left( x,\frac{\nabla u}{t}\right) -{\mathcal {A}} \left( x,\frac{\nabla v}{s}\right) + (1-\theta (x)) \left[ {\mathcal {B}}\left( x,\frac{u}{t}\right) - {\mathcal {B}}\left( x,\frac{v}{s}\right) \right] ,\nabla \varphi \right\rangle dx \\\\&\qquad = \int _\Omega \langle \theta (x) [{\mathcal {B}}(x,v)- {\mathcal {B}}(x,u)],\nabla \varphi \rangle dx \,\qquad \forall \varphi \in W^{1,2}_0(\Omega ). \end{aligned}$$
(4.9)

Then, we can use \(\varphi =\left( \frac{u}{t}-\frac{v}{s}\right) \in {\mathcal {W}}^{1,2}_0(\Omega )\) as a test function in (4.9). By (1.5) and the obvious inequality \(\vert u-t\vert \le \vert \frac{u}{t}-\frac{v}{s}\vert + \vert \frac{v}{s}\vert \) we get

$$\begin{aligned}&\left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{\mathcal L^{2}(\Omega )}^2 \\\\&\quad \leqslant \frac{1}{\lambda _0} \int _\Omega \langle \theta (x) ( {\mathcal {B}} (x,v)-{\mathcal {B}} (x,u ) ),\frac{\nabla u}{t} - \frac{\nabla v}{s}\rangle dx \\\\&\quad \leqslant cM \int _{\Omega _k} |u-v| \left| \frac{\nabla u}{t}-\frac{\nabla v}{s} \right| dx+c M \int _{\Omega \setminus \Omega _k} |u-v | \left| \frac{\nabla u}{t}-\frac{\nabla v}{s} \right| dx \\\\&\quad \leqslant c \,|\Omega _k|^{\frac{1}{N}} S_2\left\| \frac{\nabla v}{s}-\frac{\nabla u}{t}\right\| _{{\mathcal {L}}^{2}(\Omega )}^2 + c \left\| \frac{\nabla v}{s}-\frac{\nabla u}{t}\right\| _{\mathcal L^{2}(\Omega )} \left\| \frac{v}{s} \right\| _{{\mathcal {L}}^{2}(\Omega )} \\\\&\quad \quad + c\, k \left\| \frac{\nabla v}{s}-\frac{\nabla u}{t}\right\| _{{\mathcal {L}}^{2}(\Omega )} \,|\Omega |^{\frac{1}{2} }+ c\, \left\| \frac{\nabla v}{s}-\frac{\nabla u}{t}\right\| _{\mathcal L^{2}(\Omega )} \left\| \frac{v}{s}\right\| _{{\mathcal {L}}^{2}(\Omega )} \end{aligned}$$
(4.10)

where \(\Omega _k=\left\{ x\in \Omega : \left| \frac{ u}{t} - \frac{ v}{s}\right| >k\right\} \).

Dividing both sides of (4.10) by \( \left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{\mathcal L^2(\Omega )}\ \) we obtain:

$$\begin{aligned} \left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{\mathcal L^{2}(\Omega )} \leqslant c \,|\Omega _k|^{\frac{1}{N}} S_2\left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{\mathcal L^{2}(\Omega )}+ c \left( \left\| \frac{v}{s} \right\| _{\mathcal L^{2}(\Omega )} + \, k \,|\Omega |^{\frac{1}{2} }\right) . \end{aligned}$$
(4.11)

At this point our aim is to absorb, in last inequality, the first term on the right-hand side into the left-hand side, Therefore we estimate the measure \(|\Omega _k| \) of the superlevel set \(\Omega _k.\) We shall use here a technique due to Boccardo (see [3]).

We use \(\varphi :=\frac{ \frac{u}{t} -\frac{v}{s}}{ 1+|\frac{u}{t}-\frac{v}{s}|}\in {\mathcal {W}}^{1,2}_0(\Omega )\) as test function in (4.9) obtaining

$$\begin{aligned} \begin{aligned}&\int _\Omega \left\langle {\mathcal {A}} \left( x,\frac{\nabla u}{t} \right) -{\mathcal {A}} \left( x,\frac{\nabla v}{s} \right) + (1-\theta (x)) \left[ {\mathcal {B}} \left( x,{u\over t} \right) - {\mathcal {B}} \left( x,{v\over s} \right) \right] , \frac{ \nabla \left( \frac{u}{t}\right) - \nabla \left( \frac{v}{s}\right) }{ \left( 1+\vert \frac{u}{t}- \frac{v}{s} \vert \right) ^2 } \right\rangle \;dx \\\\&\quad = \int _\Omega \left\langle \theta (x) [{\mathcal {B}}(x,v)- {\mathcal {B}}(x,u)], \frac{ \nabla \left( \frac{u}{t}\right) - \nabla \left( \frac{v}{s}\right) }{ \left( 1+\vert \frac{u}{t}- \frac{v}{s} \vert \right) ^2 } \right\rangle \; dx. \end{aligned} \end{aligned}$$

On the other hand, since

$$\begin{aligned} |\theta (x)\left( {\mathcal {B}}(x,u)- \mathcal B\left( x,\frac{t}{s}v\right) \right) |\leqslant M t \left| \frac{u}{t}-\frac{v}{s} \right| \end{aligned}$$

and

$$\begin{aligned} |\theta (x) \left( {\mathcal {B}}\left( x,\frac{t}{s}v\right) -{\mathcal {B}}(x,v) \right) |\leqslant M |t-s| \left| \frac{v}{s} \right| , \end{aligned}$$

we get

$$\begin{aligned}&\left\| \frac{\nabla \frac{u}{t}-\nabla \frac{v}{s}}{ 1+\left| \frac{u}{t}-\frac{v}{s}\right| }\right\| _{{\mathcal {L}}^2(\Omega )}^2\leqslant \int _\Omega M \frac{\left| \nabla \frac{u}{t}-\nabla \frac{v}{s}\right| }{ 1+\left| \frac{u}{t}- \frac{v}{s}\right| }\; dx+ \int _\Omega M \left| \frac{v}{s} \right| \frac{\left| \nabla \frac{u}{t}-\nabla \frac{v}{s}\right| }{\left( 1+\left| \frac{u}{t}-\frac{v}{s}\right| \right) ^2} \;dx \\\\&\\\\&\quad \leqslant M \left\| \frac{ \nabla \frac{u}{t}-\nabla \frac{v}{s}}{ 1+\left| \frac{u}{t}-\frac{v}{s}\right| } \right\| _{\mathcal L^2(\Omega )} |\Omega | ^\frac{1}{2} + M \left\| \frac{ \nabla \frac{u}{t}-\nabla \frac{v}{s}}{ 1+\left| \frac{u}{t}-\frac{v}{s}\right| } \right\| _{{\mathcal {L}}^2(\Omega )} \left\| \frac{v}{s} \right\| _{\mathcal L^2(\Omega )}, \end{aligned}$$

so that

$$\begin{aligned} \left\| \frac{ \nabla (\frac{u}{t})- \nabla (\frac{v}{s}) }{ \left( 1+\left| \frac{u}{t}- \frac{v}{s} \right| \right) } \right\| _{{\mathcal {L}}^2(\Omega )} \le M \left( \vert \Omega \vert ^{\frac{1}{2}}+ \left\| {v\over s} \right\| _{\mathcal L^2(\Omega )} \right) . \end{aligned}$$
(4.12)

From (4.12) and (2.1) we have

$$\begin{aligned} \left\| \log \left( 1+\left| \frac{u}{t} - \frac{v}{s} \right| \right) \right\| _{{\mathcal {L}}^{2^*}(\Omega )}\leqslant c M S_{2} \left( \left\| \frac{v}{s} \right\| _{{\mathcal {L}}^2(\Omega )} +|\Omega | ^\frac{1}{2} \right) . \end{aligned}$$

and this gives, \(\forall k\in {\mathbb {N}},\)

$$\begin{aligned} |\Omega _k| ^{\frac{1}{2^*}}\leqslant \frac{CM }{ \log \left( 1+ k\right) } \left( \left\| \frac{v}{s} \right\| _{\mathcal L^2(\Omega )} +|\Omega | ^\frac{1}{2} \right) , \end{aligned}$$
(4.13)

where \(C=C(N, \lambda _0, \Omega )\).

At this point, in view of (4.13), in (4.11) we can choose \(k={{\bar{k}}}={{\bar{k}}}(\lambda _0,\Lambda , N,\Omega , \Vert \mathcal F\Vert _{L^2{log}^{-\alpha } {\mathcal {L}}(\Omega )},\Vert \beta \Vert _{BMO}, d)\) large enough, increasing with respect to \(\Vert \beta \Vert _{BMO}\), in such a way that

$$\begin{aligned} c|\Omega _{{{\bar{k}}}}|^\frac{1}{N} S_2 = \frac{1}{2}. \end{aligned}$$
(4.14)

Precisely, we can fix the value

$$\begin{aligned} {{\bar{k}}} = Ce^{C \Vert \frac{v}{s} \Vert _{{\mathcal {L}}^{2} (\Omega )} } -1, \end{aligned}$$
(4.15)

with C positive constant increasing with respect to \(\Vert \beta \Vert _{BMO}\). Hence, by (4.11) and (4.14) we have that

$$\begin{aligned} \left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{\mathcal L^{2}(\Omega )} \leqslant c \left( \left\| \frac{v}{s} \right\| _{{\mathcal {L}}^{2}(\Omega )} + \, {{\bar{k}}} \,|\Omega |^{\frac{1}{2} }\right) . \end{aligned}$$
(4.16)

Moreover, from (2.8) we have

$$\begin{aligned} \left\| \frac{v}{s}\right\| _{{\mathcal {L}}^{2}(\Omega )}\le c \left\| \frac{v}{s}\right\| _{{\mathcal {L}}^{2^*} {log}^{r} {\mathcal {L}}(\Omega )}\le cS \left\| \frac{\nabla v}{ s}\right\| _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} \end{aligned}$$
(4.17)

so that, combining (4.16), (4.15), (4.17) and (4.5) we arrive to

$$\begin{aligned} \left\| \frac{\nabla u}{t}-\frac{\nabla v}{s}\right\| _{\mathcal L^2(\Omega ;{{\mathbb {R}}}^{n})}\leqslant C\left( \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} + e^{ C \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} } \right) . \end{aligned}$$
(4.18)

Finally, using (4.6), (4.8), (4.5) and (4.18) we obtain

$$\begin{aligned} \Vert \nabla u\Vert _{{\mathcal {L}}^2{log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} \leqslant C\left( \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} + e^{ C \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} } \right) \end{aligned}$$
(4.19)

where \(C=C ( \lambda _0, \Lambda , N,\Omega , \Vert \beta \Vert _{BMO}, d)\) is increasing with respect to \(\Vert \beta \Vert _{BMO}\).

Applying the Leray–Shauder Fixed Point Theorem 4.1, where we choose \({\mathbb {K}} = C\left( \Vert {\mathcal {F}} \Vert _{\mathcal L^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} + e^{ C \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} } \right) \), we obtain that \({\mathbb {F}}\) has a fixed point and this completes the proof of Theorem 1.2 in case \(\beta \in {\mathcal {L}}^\infty (\Omega )\).

Step 2 Let us remove the additional assumption \(\beta \in {\mathcal {L}}^{\infty }(\Omega )\) and let us consider the more general case \(\beta \in BMO(\Omega )\).

We shall use an approximation argument. In particular we shall consider a sequence of problems of our type for operators where the principal part is a ’truncated’ \({\mathcal {A}}_n\) of \(\mathcal A\) (see (4.20) and (4.24)), for which we can apply the a priori estimate (4.19) obtained in the last step of the proof.

For every \(n\in {\mathbb {N}}\) and for almost every \(x \in \Omega \) let us consider the following operators:

$$\begin{aligned} {\mathcal {A}}_n (x,\xi ):= \left\{ \begin{array}{ll} {\mathcal {A}}(x,\xi ) &{}\quad \text{ if } \beta (x)\leqslant n \\ n\frac{{\mathcal {A}}(x,\xi )}{\beta (x)} &{}\quad \text{ if } \beta (x)> n. \end{array} \right. \end{aligned}$$
(4.20)

For every \(\xi , \eta \in {{\mathbb {R}}}^{N}\) the following properties are obviously satisfied:

$$\begin{aligned}&| {\mathcal {A}}_n (x, \xi ) - {\mathcal {A}}_n (x, \eta ) | \leqslant \Lambda T_n\beta (x) |\xi -\eta | \end{aligned}$$
(4.21)
$$\begin{aligned}&T_n \beta (x)|\xi -\eta |^2 \leqslant \langle {\mathcal {A}}_n(x, \xi )- {\mathcal {A}}_n (x, \eta ), \xi -\eta \rangle \end{aligned}$$
(4.22)
$$\begin{aligned}&{\mathcal {A}}_n(x,0)=0, \end{aligned}$$
(4.23)

where \(T_n \beta (x)\) denotes the usual truncated of \(\beta \) at level n (see (2.2)).

Noting that \(T_n \beta \in {\mathcal {L}}^\infty (\Omega )\) and that, for n sufficiently large, \(T_n \beta (x)\geqslant \lambda _0>0\), we can use the existence result obtained in last step. Hence, for every \(n\in {\mathbb {N}}\) there exists a solution \(u_n\) to the problem

$$\begin{aligned} \left\{ \begin{array}{lll} \text {div}({\mathcal {A}}_n(x,\nabla u_n) + {\mathcal {B}}(x,u_n)) &{} = {\text {div}}\;{\mathcal {F}} &{} \text {in} \,\,\,\Omega \\ &{} &{} \\ u_n\in {\mathcal {W}}^{1,2}_0 {log}^{-\alpha } {\mathcal {L}}(\Omega ) \\ \end{array} \right. \end{aligned}$$
(4.24)

and

$$\begin{aligned} ||\nabla u_n||_{ {\mathcal {L}}^{2}{log}^{-\alpha } {\mathcal {L}}(\Omega .{{\mathbb {R}}}^{n})} \leqslant C\left( \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} + e^{ C \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ;{{\mathbb {R}}}^{n})} } \right) \end{aligned}$$
(4.25)

holds, with C increasing with respect to \(\Vert T_n \beta ||_{BMO}\).

Observing that by [6] we have

$$\begin{aligned} \Vert T_n \beta \Vert _{BMO}\leqslant 2\Vert \beta \Vert _{BMO}, \end{aligned}$$
(4.26)

by (4.25) we obtain that the solution sequence \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(\mathcal W^{1,2}_0{log}^{-\alpha } {\mathcal {L}}(\Omega .{{\mathbb {R}}}^{n}) \). Then, unless to pass to a subsequence, \(\{u_n\}_{n\in {\mathbb {N}}}\) converges weakly in \({\mathcal {W}}_0^{1,2}{log}^{-\alpha } {\mathcal {L}}(\Omega )\) and strongly in \({\mathcal {L}}^2{log}^{-\alpha } {\mathcal {L}}(\Omega )\) to a function \(u\in {\mathcal {W}}_0^{1,2}{log}^{-\alpha } {\mathcal {L}}(\Omega )\).

To prove that such limit function u solves our problem (1.1), it is sufficient to prove that \(\{u_n\}_{n\in {\mathbb {N}}} \) is a Cauchy sequence in \({\mathcal {W}}^{1,r}_0( \Omega )\) for some \(r<2\).

To this aim, we recall that, by [37, Theorem 2.1] in our assumptions there exists \(\varepsilon _0=\varepsilon _0 (\alpha , \Vert \beta \Vert _{BMO}, N,\lambda _0)\) such that for every \(2-\varepsilon _0< r <2\) and every \({\mathcal {F}}\in {\mathcal {L}}^r(\Omega , {{\mathbb {R}}}^{n})\), problem (1.1) admits a unique solution \(u\in {\mathcal {W}}^{1,r}_0(\Omega )\). So, let us fix an exponent r in such range close to 2, that is \(2-\varepsilon _0<r<2.\) By (4.25) and using (2.7), it immediately follows that \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded also in \({\mathcal {W}} ^{1,p} (\Omega )\) for every \(2-\varepsilon _0<r<p<2\). At this point, we can follow line by line the proof of [37, pag. 1404]. For the sake of completeness we give here some detail.

For any \(n,m\in {\mathbb {N}}\), since \((u_n-u_m)\in \mathcal W^{1,r}_0(\beta _n,\Omega ) \) we can use the Stability of the Hodge Decomposition stated in [28, 29] (see also [10, Lemma 2]) to obtain that there exist \(\varphi \in \mathcal W_0^{1,\frac{r}{r-1}}(\beta _n,\Omega )\) and a divergence free vector field \(h\in {\mathcal {L}}^{\frac{r}{r-1}}(\beta _n,\Omega )\) such that

$$\begin{aligned} \begin{aligned}&|\nabla u_n-\nabla u_m|^{r-2}(\nabla u_n-\nabla u_m)=\nabla \varphi +h,\\&\Vert \nabla \varphi \Vert _{{\mathcal {L}}^{\frac{r}{r-1}}(\beta _n,\Omega )}\leqslant c(N) (1+\Vert \beta _n\Vert _{BMO})^\gamma \Vert \nabla u_n-\nabla u_m\Vert ^{r-1}_{{\mathcal {L}}^{r}(\beta _n,\Omega )} \\&\Vert h\Vert _{{\mathcal {L}}^{\frac{r}{r-1}}(\beta _n,\Omega )}\leqslant c(N) (1+\Vert \beta _n\Vert _{BMO})^\gamma (2-r) \Vert \nabla u_n-\nabla u_m\Vert ^{r-1}_{{\mathcal {L}}^{r}(\beta _n,\Omega )}.\\ \end{aligned} \end{aligned}$$
(4.27)

where \(\gamma \) is a universal positive constant. Without loss of generality we can assume \(m<n\). Then, by (4.20), (4.21), (4.22), (4.26) and (4.27) we have

$$\begin{aligned} \begin{aligned}&\Vert \nabla u_n-\nabla u_m \Vert _{{\mathcal {L}}^{r}(\Omega .{{\mathbb {R}}}^{n})}\\&\quad \leqslant \lambda _0^{-1} \int _{\Omega }\left\langle {\mathcal {A}}_n\left( x,\nabla u_n \right) -{\mathcal {A}}_n\left( x,\nabla u_m \right) ,\nabla u_n-\nabla u_m \right\rangle |\nabla u_n-\nabla u_m|^{r-2} \text{d}x\\&\quad =\lambda _0^{-1} \left[ \int _{\Omega }\left\langle {\mathcal {A}}_n\left( x,\nabla u_n \right) -{\mathcal {A}}_n\left( x,\nabla u_m \right) ,h \right\rangle \text{d}x \right. \\&\qquad +\, \int _{\Omega }\left\langle {\mathcal {A}}_n\left( x,\nabla u_n \right) -{\mathcal {A}}_m\left( x,\nabla u_m \right) ,\nabla \varphi \right\rangle \text{d}x\\&\qquad +\,\left. \int _{\Omega }\left\langle {\mathcal {A}}_m\left( x,\nabla u_m \right) -{\mathcal {A}}_n\left( x,\nabla u_m\right) ,\nabla \varphi \right\rangle \text{d}x \right] \\&\quad \leqslant \lambda _0^{-1} c(N)\big ( 1+2\Vert \beta \Vert _{BMO}\big )^\gamma \left[\vphantom{\Vert \beta _m-\beta _n\Vert _{ {\mathcal {L}}^{\frac{rp}{p-r}} (\Omega ) }} \Vert {\mathcal {B}}(x,u_n)-{\mathcal {B}} (x,u_m)\Vert _{{\mathcal {L}}^{r}(\Omega .{{\mathbb {R}}}^{n})} \right. \\&\qquad \left. +\,\Lambda \Vert \nabla u_m \Vert _{{\mathcal {L}}^p(\Omega .{{\mathbb {R}}}^{n})} \Vert \beta _m-\beta _n\Vert _{ {\mathcal {L}}^{\frac{rp}{p-r}} (\Omega ) } \right] .\\ \end{aligned} \end{aligned}$$
(4.28)

From (1.7), since \( u_n \rightarrow u \) a.e. on \(\Omega \) we have \({\mathcal {B}}(x, u_n) \rightarrow {\mathcal {B}}(x, u)\) a.e. in \(\Omega .\) Moreover, being \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \( \mathcal W^{1,p}_0(\Omega ) \), by assumptions (1.7)–(1.8) and Theorem 2.3 we have that \( \Vert {\mathcal {B}}(x,u_n)\Vert _{ \mathcal L^p(\Omega , {{\mathbb {R}}}^{n}) }\leqslant C\).

Since \(r<p,\) this implies \({\mathcal {B}}(x, u_n)\rightarrow \mathcal B(x, u )\) strongly in \({\mathcal {L}}^{r} (\Omega ; {{\mathbb {R}}}^{n})\). Combining these considerations with (4.28), we obtain that \(\{u_n\}_{n\in {\mathbb {N}}}\) is a Cauchy sequence in \(\mathcal W^{1,r}_0(\Omega )\) as we claimed. This ensures that \(u_{n}\) converges to a solution \(u\in {\mathcal {W}}^{1,2}_0{\mathcal {L}} \log ^{-\alpha } {\mathcal {L}} (\Omega )\) of problem (1.1) and the a priori estimate

$$\begin{aligned} ||\nabla u||_{ \mathcal L^{2}{log}^{-\alpha } {\mathcal {L}}(\Omega ,{{\mathbb {R}}}^{n})} \leqslant C\left( \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } \mathcal L(\Omega ,{{\mathbb {R}}}^{n})} + e^{ C \Vert {\mathcal {F}} \Vert _{{\mathcal {L}}^{2} {log}^{-\alpha } {\mathcal {L}}(\Omega ,{{\mathbb {R}}}^{n})} } \right) \end{aligned}$$

holds with \(C=C ( \lambda _0, \Lambda , N,\Omega , \Vert \beta \Vert _{BMO}, d)\) increasing with respect to the BMO- norm of \(\beta .\)

We conclude our proof by observing that for every \({\mathcal {F}} \in {\mathcal {L}}^2{log}^{-\alpha } {\mathcal {L}}(\Omega ) ,\) the uniqueness of solution \(u\in {\mathcal {W}}_0^{1,2}{log}^{-\alpha } {\mathcal {L}}(\Omega ) \) to problem (1.1) follows by Theorem 2.1 of [37] by using (2.7). \(\square \)