A nonlinear Dirichlet problem involving terms with logarithmic growth

Abstract

We study existence and regularity of weak solutions for a class of boundary value problems, whose form is $\begin{array}{l} \{\begin{matrix} - div (\frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u) + u | \nabla u | log (1 + | \nabla u |) = f (x), & in Ω \\ u = 0, & on \partial Ω \end{matrix} \end{array}$ where both the principal part and the lower order term have a logarithmic growth with respect to the gradient of the solutions. We prove that the solutions, due to the regularizing effect given by the lower order term, belong to the Orlicz–Sobolev space generated by the function $s log (1 + | s |)$ even for $L^{1} (Ω)$ data.

Keywords

Nonlinear partial differential equations Orlicz–Sobolev spaces Logarithmic growth

1. Introduction

In this paper we prove the existence of $W_{0}^{1, 1} (Ω)$ solutions of boundary value problems involving terms with a logarithmic growth with respect to the gradient of the solution. More precisely, we study the existence of weak solutions for the problem $\begin{array}{l} (1) & \{\begin{matrix} - div (\frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u) + u | \nabla u | log (1 + | \nabla u |) = f (x), & in Ω \\ u = 0, & on \partial Ω, \end{matrix} \end{array}$ that is $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u \nabla v + \int_{Ω} u | \nabla u | log (1 + | \nabla u |) v \\ (2) & = \int_{Ω} f v, \forall v \in W_{0}^{1, A} (Ω) \cap L^{\infty} (Ω), \end{array}$ where Ω is an open bounded subset of $R^{N}$ , with $N ⩾ 2$ , $W_{0}^{1, A} (Ω)$ is the Orlicz–Sobolev space generated by the function $\begin{array}{l} A (s) = s log (1 + s), \end{array}$ and $m (x)$ is a measurable function such that $\begin{array}{l} (3) & \exists 0 < α ⩽ β : α ⩽ m (x) ⩽ β a.e. x \in Ω . \end{array}$ We also assume that $\begin{array}{l} (4) & \begin{aligned} f \in L^{1} (Ω), \\ f (x) ⩾ 0 a.e. x \in Ω . \end{aligned} \end{array}$ The problem (1), without the lower order term, is studied in [4], where the authors prove the existence of a unique weak solution in the Orlicz–Sobolev space for the problem $\begin{array}{l} (5) & \{\begin{matrix} - div (\frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u) = f (x), & in Ω \\ u = 0, & on \partial Ω \end{matrix} \end{array}$ with $f \in L^{N} (Ω)$ . In [13] the author extends the results of [4], generalizing the growth of the principal part of (5) to ${log}^{α} (1 + | \nabla u |)$ , with $α > 0$ . Notice that the growth of the operator in the principal part is $log (1 + | \nabla u |)$ , which is bounded by the growth of any p-Laplacian, with $p > 1$ , that is $| \nabla u |^{p - 1}$ .

The presence of the lower order term in (1) is inspired by the Calculus of Variations. Indeed, if we consider the integral functional $\begin{array}{l} (6) & I (v) = \int_{Ω} a (x, v) | \nabla v | log (1 + | \nabla v |) - \int_{Ω} f v, \end{array}$ then, formally, the Euler–Lagrange equation for minima of I is $\begin{array}{l} \int_{Ω} a (x, u) \frac{log (1 + | \nabla u |)}{| \nabla u |} \nabla u \nabla v + \int_{Ω} a^{'} (x, u) | \nabla u | log (1 + | \nabla u |) v \\ (7) & + \int_{Ω} a (x, u) \frac{\nabla u}{1 + | \nabla u |} \nabla v = \int_{Ω} f v . \end{array}$ The existence of a minimizer u of I is discussed in Appendix B.

This connection with the Calculus of Variations can be found in many results about the existence of weak solutions for Dirichlet problems having natural growth lower order terms (see for example [1]). Note that the properties of the Euler–Lagrange equation of $\begin{matrix} I_{p} (v) = \int_{Ω} a (x, v) | \nabla v |^{p} - \int_{Ω} f v \end{matrix}$ were one of the inspirations to the study of nonlinear elliptic equations with lower order term having p-growth with respect to the gradient, while the growth of the principal part is $p - 1$ . The framework of (7) is slightly similar to the Euler–Lagrange equation of $I_{p}$ . Indeed, the growth of the principal part is $log (1 + | \nabla u |)$ while the growth of the lower order term is $| \nabla u | log (1 + | \nabla u |)$ .

The study of variational integrals having nearly linear growth in $| \nabla v |$ , such as (6), can be found, for example, in [5,10] and [12].

Existence results for problems with the same structure of (1) have been studied in [8] and [9].

The presence of a lower order term of the form $u log (1 + | \nabla u |) | \nabla u |$ produces in (1) a regularizing effect (as in problems motivated by the Euler–Lagrange equations of $I_{p}$ ) and we can prove the existence of weak solutions even for $L^{1} (Ω)$ data (as in [2]), as described in the following theorem.

Theorem 1.1.
Assume ( 3 ) and ( 4 ). Then there exists a nonnegative weak solution $u \in W_{0}^{1, A} (Ω)$ of the problem ( 2 ).

Furthermore, the lower order term is well defined, that is, $\begin{array}{l} u | \nabla u | log (1 + | \nabla u |) \in L^{1} (Ω) . \end{array}$

In the last part of this paper we assume a more regular datum, that is $\begin{array}{l} (8) & f \in L^{N} (Ω) \end{array}$ and thanks to the Stampacchia method (see [14]), with the same approach of [4], we will prove the following result.
Corollary 1.2.
Assume ( 8 ). Then every solution of ( 1 ) belongs to $L^{\infty} (Ω)$ .

In Appendix A we collect some results about Orlicz–Sobolev spaces to which we will often refer.
2. Proof of the main result

Consider the following approximation of the boundary value problem (2) $\begin{array}{l} \frac{1}{n} \int_{Ω} \nabla u \nabla v + \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u \nabla v + \int_{Ω} T_{n^{3}} (u) | \nabla u | log (1 + | \nabla u |) v \\ (9) & = \int_{Ω} f_{n} v, \forall v \in W_{0}^{1, 2} (Ω) \cap L^{\infty} (Ω), \end{array}$ where $\begin{array}{l} f_{n} = \frac{f}{1 + \frac{1}{n} f} \end{array}$ and we recall that $T_{k} (s)$ is the real valued function defined as $\begin{array}{l} T_{k} (s) = \{\begin{matrix} s, & | s | < k, \\ k sgn (s), & | s | ⩾ k . \end{matrix} \end{array}$ Due to a classical result of Leray and Lions (see [11]), for each n, there exists a weak solution $u_{n} \in W_{0}^{1, 2} (Ω)$ of problem (9). Furthermore, thanks to the results of Stampacchia (see [14]), $u_{n}$ belongs to $L^{\infty} (Ω)$ , although not uniformly in n. More precisely, only using the term with $\frac{1}{n} \int_{Ω} \nabla u_{n} \nabla v$ , one can prove that there exists a constant $C > 0$ such that $\begin{array}{l} ‖ u_{n} ‖_{\infty} ⩽ C n ‖ f_{n} ‖_{\infty} = C n^{2} \end{array}$ and then there exists $\overline{n} > 0$ such that $\begin{array}{l} T_{n^{3}} (u_{n}) = u_{n} \forall n ⩾ \overline{n} . \end{array}$ This allows us to write $u_{n}$ in place of $T_{n^{3}} (u_{n})$ here and everywhere.

Lemma 2.1.
Assume ( 4 ), then $u_{n} (x) ⩾ 0$ a.e.
Proof.
Consider as test function in (9) $v = u_{n}^{-}$ . $\begin{array}{l} - \frac{1}{n} \int_{Ω} {| \nabla u_{n}^{-} |}^{2} + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla u_{n}^{-} \\ + \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) u_{n}^{-} = \int_{Ω} f_{n} u_{n}^{-} ⩾ 0 . \end{array}$ In the principal part we can use (3) to obtain $\begin{array}{l} - \frac{1}{n} \int_{Ω} {| \nabla u_{n}^{-} |}^{2} - α \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} {| \nabla u_{n}^{-} |}^{2} - \int_{Ω} {| u_{n}^{-} |}^{2} | \nabla u_{n} | log (1 + | \nabla u_{n} |) ⩾ 0 \end{array}$ that implies $\begin{array}{l} \int_{Ω} {| \nabla u_{n}^{-} |}^{2} ⩽ 0, \end{array}$ otherwise said $u_{n} (x) ⩾ 0$ a.e. □

We want now estimate the norm of $u_{n}$ in $W_{0}^{1, 1} (Ω)$ .
Lemma 2.2.
The sequence ${u_{n}}$ is bounded in $W_{0}^{1, 1} (Ω)$ . Furthermore, there exists a constant $C (α)$ that depends only on α such that $\begin{array}{l} (10) & \int_{Ω} | \nabla u_{n} | log (1 + | \nabla u_{n} |) ⩽ C (α) ‖ f ‖_{1}, \end{array}$ that is $u_{n} \in W_{0}^{1, A} (Ω)$ .
Proof.
We take as test function in (9) $v = T_{j} (u_{n})$ . $\begin{array}{l} \frac{1}{n} \int_{Ω} {| \nabla T_{j} (u_{n}) |}^{2} + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla T_{j} (u_{n}) \\ + \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) T_{j} (u_{n}) ⩽ j ‖ f ‖_{1} . \end{array}$ We drop by positivity the first term and we get a lower bound if in the second term we consider the integral over ${u_{n} < j}$ and in the third term we consider the integral over ${u_{n} ⩾ j}$ (note that $T_{j} (u_{n}) u_{n} ⩾ j^{2}$ on the set ${u_{n} ⩾ j}$ ): $\begin{array}{l} α \int_{{u_{n} < j}} log (1 + | \nabla u_{n} |) | \nabla u_{n} | + j^{2} \int_{{u_{n} ⩾ j}} log (1 + | \nabla u_{n} |) | \nabla u_{n} | ⩽ j ‖ f ‖_{1} \end{array}$ hence $\begin{array}{l} \{\begin{matrix} \int_{{u_{n} < j}} log (1 + | \nabla u_{n} |) | \nabla u_{n} | ⩽ \frac{j}{α} ‖ f ‖_{1}, \\ \int_{{u_{n} ⩾ j}} log (1 + | \nabla u_{n} |) | \nabla u_{n} | ⩽ \frac{1}{j} ‖ f ‖_{1} . \end{matrix} \end{array}$ Thanks to the previous estimate we get $\begin{array}{l} \int_{Ω} log (1 + | \nabla u_{n} |) | \nabla u_{n} | \\ = \int_{{u_{n} < j}} log (1 + | \nabla u_{n} |) | \nabla u_{n} | \\ + \int_{{u_{n} ⩾ j}} log (1 + | \nabla u_{n} |) | \nabla u_{n} | ⩽ (\frac{j}{α} + \frac{1}{j}) ‖ f ‖_{1}, \end{array}$ which implies that $u_{n} \in W_{0}^{1, A} (Ω)$ (uniformly with respect to $n \in N$ ).

Let us take $B > 0$ , then $\begin{array}{l} \int_{Ω} log (1 + | \nabla u_{n} |) | \nabla u_{n} | \\ ⩾ \int_{{| \nabla u_{n} | ⩾ B}} log (1 + | \nabla u_{n} |) | \nabla u_{n} | \\ ⩾ log (1 + B) \int_{{| \nabla u_{n} | ⩾ B}} | \nabla u_{n} |, \end{array}$ that implies $\begin{array}{l} \int_{{| \nabla u_{n} | ⩾ B}} | \nabla u_{n} | & ⩽ \frac{1}{log (1 + B)} \int_{Ω} log (1 + | \nabla u_{n} |) | \nabla u_{n} | \\ (11) & ⩽ \frac{1}{log (1 + B)} (\frac{j}{α} + \frac{1}{j}) ‖ f ‖_{1} . \end{array}$ Finally, decomposing $Ω = {| \nabla u_{n} | ⩾ B} \cup {| \nabla u_{n} | < B}$ and using (11), we can conclude the proof $\begin{array}{l} \int_{Ω} | \nabla u_{n} | & = \int_{{| \nabla u_{n} | ⩾ B}} | \nabla u_{n} | + \int_{{| \nabla u_{n} | < B}} | \nabla u_{n} | ⩽ \\ ⩽ \frac{1}{log (1 + B)} \int_{Ω} log (1 + | \nabla u_{n} |) | \nabla u_{n} | + B | Ω | ⩽ \\ ⩽ \frac{1}{log (1 + B)} (\frac{j}{α} + \frac{1}{j}) ‖ f ‖_{1} + B | Ω | . \end{array}$ □

As a consequence of Lemma 2.2 and of Sobolev’s embedding theorem there exists a function u, right now only belonging to the space $B V (Ω)$ , and a subsequence $u_{n_{k}}$ (still denoted as $u_{n}$ for simplicity) such that $\begin{array}{l} u_{n} \to u in L^{1} (Ω) and a.e. \end{array}$ Now, we want to prove that the term involving the lower order term in (2) is well defined, which is the content of the next result
Lemma 2.3.
The sequence ${u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |)}$ is bounded in $L^{1} (Ω)$ .
Proof.
We consider as test function in (9) $v = \frac{T_{j} (G_{k} (u_{n}))}{j}$ , where $G_{k} (s) = s - T_{k} (s)$ for each $s \in R$ . We can drop by positivity the principal parts and we observe that $G_{k} (u_{n}) = 0$ in the region where $u_{n} ⩽ k$ , so that $\begin{array}{l} \frac{1}{j} \int_{{u_{n} ⩾ k}} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) T_{j} (G_{k} (u_{n})) ⩽ \int_{{u_{n} ⩾ k}} f . \end{array}$ We can now use Fatou’s lemma for $j \to 0$ and, since $G_{k} (u_{n}) ⩾ 0$ , we obtain $\begin{array}{l} (12) & \int_{{u_{n} ⩾ k}} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) ⩽ \int_{{u_{n} ⩾ k}} f . \end{array}$ If we take $k = 0$ , we have $\begin{array}{l} (13) & \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) ⩽ \int_{Ω} f, \end{array}$ as desired. □

In order to prove that $u_{n}$ converges weakly in $W_{0}^{1, 1} (Ω)$ , so that u will belong to $W_{0}^{1, 1} (Ω)$ and not only to $B V (Ω)$ , we check the assumptions of Dunford–Pettis’ theorem. Let E be a measurable set contained in Ω and let $B > 0$ ; then $\begin{array}{l} \int_{E} | \frac{\partial u_{n}}{\partial x_{i}} | ⩽ \int_{E} | \nabla u_{n} | ⩽ \int_{{| \nabla u_{n} | ⩾ B} \cap E} | \nabla u_{n} | + \int_{{| \nabla u_{n} | < B} \cap E} B . \end{array}$ Recalling (11), we have $\begin{array}{l} \int_{E} | \frac{\partial u_{n}}{\partial x_{i}} | ⩽ \frac{1}{log (1 + B)} (\frac{j}{α} + \frac{1}{j}) ‖ f ‖_{1} + B | E | . \end{array}$ Let $j = 1$ . For each $ε > 0$ there exists $B_{ε} > 0$ such that $\begin{array}{l} \frac{1}{log (1 + B)} (\frac{1}{α} + 1) ‖ f ‖_{1} < \frac{ε}{2}, \forall B ⩾ B_{ε} . \end{array}$ Take now $B = B_{ε}$ and take E: $| E | ⩽ \frac{ε}{2 B_{ε}}$ so that $\begin{array}{l} \int_{E} | \frac{\partial u_{n}}{\partial x_{i}} | ⩽ \int_{E} | \nabla u_{n} | < ε . \end{array}$ Due to Dunford–Pettis’ theorem there exists $y_{i} \in L^{1} (Ω)$ such that $\begin{array}{l} \frac{\partial u_{n}}{\partial x_{i}} ⇀ y_{i}, in L^{1} (Ω) \end{array}$ in other words $\begin{array}{l} \int_{Ω} \frac{\partial u_{n}}{\partial x_{i}} ϕ \to \int_{Ω} y_{i} ϕ \forall ϕ \in C_{0}^{\infty} (Ω) . \end{array}$ From the definition of weak derivative it follows that $\begin{array}{l} \int_{Ω} y_{i} ϕ = - \int_{Ω} u \frac{\partial ϕ}{\partial x_{i}}, \forall ϕ \in C_{0}^{\infty} (Ω) \end{array}$ so that $\begin{array}{l} y_{i} = \frac{\partial u}{\partial x_{i}} . \end{array}$ In conclusion $\begin{array}{l} (14) & \{\begin{matrix} u \in W_{0}^{1, 1} (Ω), \\ u_{n} ⇀ u in W_{0}^{1, 1} (Ω) . \end{matrix} \end{array}$ In order to pass to the limit in (9) we need the following result.
Lemma 2.4.
The sequence ${\nabla u_{n} (x)}$ converges to $\nabla u (x)$ a.e.
Proof.
From (10), observing that the function $A (s) = s log (1 + s)$ is convex, we can pass to the liminf by weak lower semicontinuity (see [6]) and obtain $\begin{array}{l} (15) & \int_{Ω} log (1 + | \nabla u |) | \nabla u | ⩽ C (α) ‖ f ‖_{1}, \end{array}$ that is $u \in W_{0}^{1, A} (Ω)$ . By density we can find a sequence of functions ${U_{k}}$ in $C_{0}^{\infty} (Ω)$ such that the following estimates hold true: $\begin{array}{l} (16) & ‖ U_{k} - u ‖_{W_{0}^{1, 1} (Ω)} ⩽ \frac{1}{k}, \end{array}$ and $\begin{array}{l} (17) & {‖ \nabla (U_{k} - u) ‖}_{L^{A} (Ω)} ⩽ \frac{1}{k} . \end{array}$ We take now $v = T_{j} (u_{n} - U_{k})$ as test function in (9) to obtain $\begin{array}{l} \frac{1}{n} \int_{Ω} \nabla u_{n} \nabla T_{j} (u_{n} - U_{k}) + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla T_{j} (u_{n} - U_{k}) \\ + \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) T_{j} (u_{n} - U_{k}) = \int_{Ω} f T_{j} (u_{n} - U_{k}) . \end{array}$ In the first term we write $\nabla u_{n} = \nabla (u_{n} - U_{k}) + \nabla U_{k}$ $\begin{array}{l} \frac{1}{n} \int_{Ω} \nabla (u_{n} - U_{k}) \nabla T_{j} (u_{n} - U_{k}) + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla T_{j} (u_{n} - U_{k}) \\ + \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) T_{j} (u_{n} - U_{k}) \\ = \int_{Ω} f T_{j} (u_{n} - U_{k}) - \frac{1}{n} \int_{Ω} \nabla U_{k} \nabla T_{j} (u_{n} - U_{k}) \end{array}$ Here and in the following we will denote as $ε_{n}$ any quantity that converges to zero as n tends to infinity. Now, drop by positivity the first term and observe that $\begin{array}{l} - \frac{1}{n} \int_{Ω} \nabla U_{k} \nabla T_{j} (u_{n} - U_{k}) = ε_{n} \end{array}$ then $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla T_{j} (u_{n} - U_{k}) \\ ⩽ ε_{n} + j ‖ f ‖_{1} + j \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) \\ (18) & ⩽ ε_{n} + 2 j ‖ f ‖_{1}, \end{array}$ where in the last inequality we used (13). On the left hand side we add and subtract $\frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} \nabla U_{k}$ to have $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla T_{j} (u_{n} - U_{k}) \\ = \int_{Ω} (\frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} - \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k}) \nabla T_{j} (u_{n} - U_{k}) \\ + \int_{Ω} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k} \nabla T_{j} (u_{n} - U_{k}) \end{array}$ and, denoting with $I_{j} = {| u_{n} - U_{k} | < j}$ , we have $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla T_{j} (u_{n} - U_{k}) \\ = \int_{Ω} (\frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} - \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k}) \nabla T_{j} (u_{n} - U_{k}) \\ + \int_{I_{j}} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k} \nabla (u_{n} - u) \\ + \int_{I_{j}} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k} \nabla (u - U_{k}) . \end{array}$ Let us define with $1_{I_{j}}$ the indicator function of $I_{j}$ , and use the previous equality to rewrite the left hand side of (18): $\begin{array}{l} \int_{Ω} (\frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} - \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k}) \nabla T_{j} (u_{n} - U_{k}) \\ + \int_{Ω} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} 1_{I_{j}} m (x) \nabla U_{k} \nabla (u_{n} - u) \\ + \int_{Ω} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} 1_{I_{j}} m (x) \nabla U_{k} \nabla (u - U_{k}) \\ ⩽ ε_{n} + 2 j ‖ f ‖_{1} . \end{array}$ We bring on the right hand side the third term and notice that $\begin{array}{l} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} 1_{I_{j}} m (x) \nabla U_{k} \nabla (u - U_{k}) ⩽ β log (1 + | \nabla U_{k} |) | \nabla (u - U_{k}) | \end{array}$ and then $\begin{array}{l} \int_{Ω} (\frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} - \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k}) \nabla T_{j} (u_{n} - U_{k}) \\ + \int_{Ω} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} 1_{I_{j}} m (x) \nabla U_{k} \nabla (u_{n} - u) \\ ⩽ ε_{n} + 2 j ‖ f ‖_{1} + β \int_{Ω} log (1 + | \nabla U_{k} |) | \nabla (u - U_{k}) | . \end{array}$ We use (38) (see Appendix A) on the last term $\begin{array}{l} \int_{Ω} (\frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} - \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k}) \nabla T_{j} (u_{n} - U_{k}) \\ + \int_{Ω} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} 1_{I_{j}} m (x) \nabla U_{k} \nabla (u_{n} - u) \\ ⩽ ε_{n} + 2 j ‖ f ‖_{1} + 2 β ‖ log (1 + | \nabla U_{k} |) ‖_{L^{\tilde{A}} (Ω)} ‖ ‖ \nabla (u - U_{k}) ‖_{L^{A} (Ω)} \end{array}$ and, due to (17) and (39) (see again Appendix A), we get $\begin{array}{l} \int_{Ω} (\frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} - \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k}) \nabla T_{j} (u_{n} - U_{k}) \\ + \int_{Ω} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} 1_{I_{j}} m (x) \nabla U_{k} \nabla (u_{n} - u) \\ ⩽ ε_{n} + 2 (j ‖ f ‖_{1} + \frac{β}{k} max {1, ‖ \nabla U_{k} ‖_{1}}) . \end{array}$ In the second term we notice that $\begin{array}{l} | \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} 1_{I_{h}} m (x) \nabla U_{k} | ⩽ β | log (1 + | \nabla U_{k} |) | \in L^{\infty} (Ω) \end{array}$ and the weak convergence of $u_{n}$ to u implies that $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} 1_{I_{j}} m (x) \nabla U_{k} \nabla (u_{n} - u) = ε_{n} \end{array}$ and then $\begin{array}{l} \int_{Ω} (\frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} - \frac{log (1 + | \nabla U_{k} |)}{| \nabla U_{k} |} m (x) \nabla U_{k}) \nabla T_{j} (u_{n} - U_{k}) \\ (19) & ⩽ ε_{n} + 2 (j ‖ f ‖_{1} + \frac{β}{k} max {1, ‖ \nabla U_{k} ‖_{1}}) . \end{array}$ We will now use the following result, whose proof can be found in [13]. Lemma 2.5.
Let g be the vector function on $R^{N}$ defined by $\begin{array}{l} g (x, ξ) = \frac{log (1 + | ξ |)}{| ξ |} m (x) ξ \end{array}$ then $\begin{array}{l} (g (x, ξ) - g (x, η)) (ξ - η) ⩾ \frac{α}{2} \frac{| ξ - η |^{2}}{1 + | ξ | + | η |} . \end{array}$

Using the previous result on the left hand side of (19) we get $\begin{array}{l} \frac{α}{2} \int_{Ω} \frac{| \nabla (u_{n} - U_{k}) |^{2}}{1 + | \nabla u_{n} | + | \nabla U_{k} |} 1_{{| u_{n} - U_{k} | ⩽ h}} ⩽ ε_{n} + 2 (j ‖ f ‖_{1} + \frac{β}{k} max {1, ‖ \nabla U_{k} ‖_{1}}) . \end{array}$ that is $\begin{array}{l} \frac{α}{2} \int_{Ω} \frac{| \nabla T_{j} (u_{n} - U_{k}) |^{2}}{1 + | \nabla u_{n} | + | \nabla U_{k} |} ⩽ ε_{n} + 2 (j ‖ f ‖_{1} + \frac{β}{k} max {1, ‖ \nabla U_{k} ‖_{1}}) . \end{array}$ Now, we estimate the $L^{1} (Ω)$ norm of $\nabla T_{j} (u_{n} - U_{k})$ : $\begin{array}{l} \frac{α}{2} \int_{Ω} | \nabla T_{j} (u_{n} - U_{k}) | \\ = \frac{α}{2} \int_{Ω} \frac{| \nabla T_{j} (u_{n} - U_{k}) |}{\sqrt{1 + | \nabla u_{n} | + | \nabla U_{k} |}} \sqrt{1 + | \nabla u_{n} | + | \nabla U_{k} |} \\ ⩽ {(\frac{α}{2} \int_{Ω} \frac{| \nabla T_{j} (u_{n} - U_{k}) |^{2}}{1 + | \nabla u_{n} | + | \nabla U_{k} |})}^{\frac{1}{2}} {(\frac{α}{2} \int_{Ω} (1 + | \nabla u_{n} | + | \nabla U_{k} |))}^{\frac{1}{2}} \\ ⩽ {(ε_{n} + 2 (j ‖ f ‖_{1} + \frac{β}{k} max {1, ‖ \nabla U_{k} ‖_{1}}))}^{\frac{1}{2}} {(\frac{α}{2} \int_{Ω} (1 + | \nabla u_{n} | + | \nabla U_{k} |))}^{\frac{1}{2}} \end{array}$ and we observe that the last term is bounded both in n and in k, hence there exists $\tilde{C} > 0$ such that $\begin{array}{l} {(\frac{α}{2} \int_{Ω} (1 + | \nabla u_{n} | + | \nabla U_{k} |))}^{\frac{1}{2}} ⩽ \tilde{C} \end{array}$ and then $\begin{array}{l} \int_{Ω} | \nabla T_{j} (u_{n} - U_{k}) | ⩽ \frac{2}{α} \tilde{C} {(ε_{n} + 2 (j ‖ f ‖_{1} + \frac{β}{k} max {1, ‖ \nabla U_{k} ‖_{1}}))}^{\frac{1}{2}} . \end{array}$ We can now use Fatou’s lemma for $k \to \infty$ and get $\begin{array}{l} (20) & \int_{Ω} | \nabla T_{j} (u_{n} - u) | ⩽ \frac{2}{α} \tilde{C} {(ε_{n} + 2 j ‖ f ‖_{1})}^{\frac{1}{2}} . \end{array}$ We can conclude the proof thanks to (20) and Lemma 2.2 and Hölder’s inequality $\begin{array}{l} \int_{Ω} {| \nabla (u_{n} - u) |}^{\frac{1}{2}} \\ = \int_{Ω} {| \nabla T_{j} (u_{n} - u) |}^{\frac{1}{2}} + \int_{{j < | u_{n} - u |}} {| \nabla (u_{n} - u) |}^{\frac{1}{2}} \\ ⩽ {(\int_{Ω} | \nabla T_{j} (u_{n} - u) |)}^{\frac{1}{2}} | Ω |^{\frac{1}{2}} + \int_{Ω} {| \nabla (u_{n} - u) |}^{\frac{1}{2}} 1_{{j < | u_{n} - u |}} \\ ⩽ {(\frac{2}{α} \tilde{C} {(ε_{n} + 2 j ‖ f ‖_{1})}^{\frac{1}{2}})}^{\frac{1}{2}} | Ω |^{\frac{1}{2}} + {(\int_{Ω} | \nabla u_{n} | + \int_{Ω} | \nabla u |)}^{\frac{1}{2}} {| {j < | u_{n} - u |} |}^{\frac{1}{2}} . \end{array}$ We pass to the lim sup and get, for each $j > 0$ $\begin{array}{l} \underset{n \to \infty}{lim sup} \int_{Ω} {| \nabla (u_{n} - u) |}^{\frac{1}{2}} ⩽ {(\frac{2}{α} \tilde{C})}^{\frac{1}{2}} {(2 j ‖ f ‖_{1})}^{\frac{1}{4}} | Ω |^{\frac{1}{2}}, \end{array}$ since $u_{n}$ converges to u in measure. Hence, taking j arbitrarily small, $\begin{array}{l} lim_{n \to \infty} \int_{Ω} {| \nabla (u_{n} - u) |}^{\frac{1}{2}} = 0 \end{array}$ that implies (up to subsequences) that $\nabla u_{n} (x) \to \nabla u (x)$ as $n \to \infty$ . □

Thanks to Lemma 2.4 and to the equiintegrability proven in Lemma 2.2, due to Vitali’s theorem, we have $\begin{array}{l} (21) & \nabla u_{n} \to \nabla u in {(L^{1} (Ω))}^{N} . \end{array}$ Now, in order to prove Theorem 1.1, we will use the “double Fatou” technique (see for example [1]). We choose as test function in (9) $0 ⩽ ϕ \in W_{0}^{1, \infty} (Ω)$ to have $\begin{array}{l} \frac{1}{n} \int_{Ω} \nabla u_{n} \nabla ϕ + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla ϕ + \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) ϕ = \int_{Ω} f_{n} ϕ . \end{array}$ Since $u_{n} ϕ ⩾ 0$ , and using (21), we can apply Fatou’s lemma and conclude that $\begin{array}{l} (22) & \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u \nabla ϕ + \int_{Ω} u | \nabla u | log (1 + | \nabla u |) ϕ ⩽ \int_{Ω} f ϕ . \end{array}$ We need now the opposite inequality and to achieve this we consider the real value function $H (t) = e^{\frac{1}{2} λ t^{2}}$ , with $λ > 0$ , and as test function $\begin{array}{l} v = \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ, \end{array}$ where $0 ⩽ ϕ \in W_{0}^{1, \infty} (Ω)$ and $U_{k}$ is as in Lemma 2.4. Observing that $\begin{array}{l} (23) & H^{'} (t) = λ t H (t) \end{array}$ and $\begin{array}{l} (24) & {[\frac{1}{H (t)}]}^{'} = - λ t \frac{1}{H (t)} \end{array}$ we have $\begin{array}{l} \frac{1}{n} \int_{Ω} λ T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \nabla u_{n} \nabla T_{j} (U_{k}) - \frac{1}{n} \int_{Ω} λ T_{j} (u_{n}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} {| \nabla T_{j} (u_{n}) |}^{2} \\ + \frac{1}{n} \int_{Ω} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} \nabla u_{n} \nabla ϕ + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) λ T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \nabla u_{n} \nabla T_{j} (U_{k}) \\ - \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) λ T_{j} (u_{n}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ {| \nabla T_{j} (u_{n}) |}^{2} \\ + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} \nabla u_{n} \nabla ϕ + \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \\ (25) & = \int_{Ω} f_{n} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ . \end{array}$ Since $\begin{array}{l} - \frac{1}{n} \int_{Ω} λ T_{j} (u_{n}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} {| \nabla T_{j} (u_{n}) |}^{2} ⩽ 0 \end{array}$ we can drop the second term. Furthermore, we observe that $\begin{array}{l} \frac{1}{n} \int_{Ω} λ T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \nabla u_{n} \nabla T_{j} (U_{k}) = ε_{n} \end{array}$ and $\begin{array}{l} \frac{1}{n} \int_{Ω} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} \nabla u_{n} \nabla ϕ = ε_{n} \end{array}$ hence (25) becomes $\begin{array}{l} ε_{n} + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) λ T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \nabla u_{n} \nabla T_{j} (U_{k}) \\ - \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) λ T_{j} (u_{n}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ {| \nabla T_{j} (u_{n}) |}^{2} \\ + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} \nabla u_{n} \nabla ϕ + \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \\ (26) & ⩾ \int_{Ω} f_{n} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ . \end{array}$ Moreover, on the left hand side we can split the lower order term $\begin{array}{l} \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} \\ = \int_{Ω} T_{j} (u_{n}) | \nabla T_{j} (u_{n}) | log (1 + | \nabla T_{j} (u_{n}) |) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \\ + \int_{Ω} u_{n} | \nabla G_{j} (u_{n}) | log (1 + | \nabla G_{j} (u_{n}) |) \frac{H (T_{j} (U_{k}))}{H (j)} ϕ \\ (27) & ⩽ \int_{Ω} T_{j} (u_{n}) | \nabla T_{j} (u_{n}) | log (1 + | \nabla T_{j} (u_{n}) |) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ + ‖ ϕ ‖_{\infty} \int_{{u_{n} ⩾ j}} f, \end{array}$ where in the last inequality we used (12). Putting together (26) and (27) we have $\begin{array}{l} ε_{n} + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) λ T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \nabla u_{n} \nabla T_{j} (U_{k}) \\ - λ \int_{Ω} log (1 + | \nabla T_{j} (u_{n}) |) m (x) T_{j} (u_{n}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ | \nabla T_{j} (u_{n}) | \\ + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} \nabla u_{n} \nabla ϕ \\ + \int_{Ω} T_{j} (u_{n}) | \nabla T_{j} (u_{n}) | log (1 + | \nabla T_{j} (u_{n}) |) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \\ (28) & + ‖ ϕ ‖_{\infty} \int_{{u_{n} ⩾ j}} f ⩾ \int_{Ω} f_{n} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ . \end{array}$ Moreover, $\begin{array}{l} lim_{j \to \infty} \int_{{u_{n} ⩾ j}} f = 0, \end{array}$ so, for the sake of simplicity, we will denote $ω_{j} = ‖ ϕ ‖_{\infty} \int_{{u_{n} ⩾ j}} f$ and then $\begin{array}{l} ε_{n} + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) λ T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ \nabla u_{n} \nabla T_{j} (U_{k}) \\ - \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} \nabla u_{n} \nabla ϕ + ω_{j} \\ ⩾ \int_{Ω} f_{n} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ + λ \int_{Ω} log (1 + | \nabla T_{j} (u_{n}) |) m (x) T_{j} (u_{n}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ | \nabla T_{j} (u_{n}) | \\ (29) & - \int_{Ω} T_{j} (u_{n}) | \nabla T_{j} (u_{n}) | log (1 + | \nabla T_{j} (u_{n}) |) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ . \end{array}$ In order to conclude the proof we will pass to the limit first in n, then in k and finally in j. Note that $\begin{array}{l} \frac{1}{H (T_{j} (u_{n}))} ⩽ 1 . \end{array}$ We start from the right hand side: since $\begin{array}{l} f_{n} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} ϕ ⩽ e^{\frac{1}{2} λ j^{2}} ‖ ϕ ‖_{\infty} f \in L^{1} (Ω) \end{array}$ we can use the dominated convergence theorem to pass to the limit. Now, consider the function $\begin{array}{l} y_{n} = \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) λ T_{j} (U_{k}) 1_{{| U_{k} | ⩽ j}} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u_{n}))} \nabla u_{n} \nabla U_{k} . \end{array}$ Then $y_{n}$ converges almost everywhere to $\begin{array}{l} (30) & y = \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) λ T_{j} (U_{k}) 1_{{| U_{k} | ⩽ j}} \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} \nabla u \nabla U_{k} . \end{array}$ Note now that $\begin{array}{l} | y_{n} | ⩽ λ β j e^{\frac{1}{2} λ j^{2}} | \nabla u_{n} ‖ \nabla U_{k} | \end{array}$ and due to the strong convergence of $u_{n}$ in $W_{0}^{1, 1} (Ω)$ we can pass to the limit in n thanks to the generalized form of the dominated convergence theorem (recall that $\nabla U_{k} \in {(L^{\infty} (Ω))}^{N}$ ). With the same idea we can pass to the limit in the second integral of (29). On the right hand side, if we suppose $λ > \frac{1}{α}$ , we can use Fatou’s lemma and we finally get $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) λ T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} \nabla u \nabla T_{j} (U_{k}) \\ + \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} \nabla u \nabla ϕ + ω_{j} \\ ⩾ \int_{Ω} f \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} ϕ + λ \int_{Ω} log (1 + | \nabla T_{j} (u) |) m (x) T_{j} (u) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} ϕ | \nabla T_{j} (u) | \\ (31) & - \int_{Ω} T_{j} (u) | \nabla T_{j} (u) | log (1 + | \nabla T_{j} (u) |) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} ϕ . \end{array}$ In order to pass to the limit in k in the first term of (31) we want to prove that $\begin{array}{l} lim_{k \to \infty} | \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) λ (T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} \nabla u \nabla T_{j} (U_{k}) - T_{j} (u) \nabla u \nabla T_{j} (u)) | = 0 \end{array}$ and to achieve this we use that $\nabla T_{j} (U_{k}) = \nabla U_{k} 1_{{| U_{k} | ⩽ j}}$ $\begin{array}{l} | \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) λ (T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} \nabla u \nabla T_{j} (U_{k}) - T_{j} (u) \nabla u \nabla T_{j} (u)) | \\ ⩽ β λ \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} | T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} 1_{{| U_{k} | ⩽ j}} \nabla u \nabla U_{k} - T_{j} (u) 1_{{| u | ⩽ j}} | \nabla u |^{2} | \\ ⩽ β λ \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} | T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} 1_{{| U_{k} | ⩽ j}} \nabla u \nabla U_{k} \\ - T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} 1_{{| U_{k} | ⩽ j}} | \nabla u |^{2} | \\ (32) & + β λ \int_{Ω} log (1 + | \nabla u |) | \nabla u | | T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} 1_{{| U_{k} | ⩽ j}} - T_{j} (u) 1_{{| u | ⩽ j}} | = I_{1} + I_{2} . \end{array}$ We observe now that $\begin{array}{l} I_{1} = & β λ \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} | T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} 1_{{| U_{k} | ⩽ j}} \nabla u \nabla U_{k} \\ - T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} 1_{{| U_{k} | ⩽ j}} | \nabla u |^{2} | \\ ⩽ & β λ \int_{Ω} log (1 + | \nabla u |) T_{j} (U_{k}) \frac{H (T_{j} (U_{k}))}{H (T_{j} (u))} 1_{{| U_{k} | ⩽ j}} | \nabla (U_{k} - u) | \end{array}$ and then, by the definition of $H (t)$ , $\begin{array}{l} I_{1} & ⩽ β λ j e^{\frac{1}{2} λ j^{2}} \int_{Ω} log (1 + | \nabla u |) | \nabla (U_{k} - u) | ⩽ 2 β λ j e^{\frac{1}{2} λ j^{2}} max {1, ‖ \nabla u ‖_{1}} {‖ \nabla (U_{k} - u) ‖}_{L^{A} (Ω)} \\ ⩽ 2 β λ j e^{\frac{1}{2} λ j^{2}} max {1, ‖ \nabla u ‖_{1}} \frac{1}{k} \end{array}$ due to (38), (39) and (17). Hence $\begin{array}{l} lim_{k \to \infty} I_{1} = 0 . \end{array}$ We can pass to the limit on $I_{2}$ thanks to the dominated convergence theorem and then $\begin{array}{l} lim_{k \to \infty} I_{2} = 0 . \end{array}$ Applying again Fatou’s lemma on the right hand side we get $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) λ T_{j} (u) \nabla u \nabla T_{j} (u) + \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u \nabla ϕ + ω_{j} \\ ⩾ \int_{Ω} f ϕ + λ \int_{Ω} log (1 + | \nabla T_{j} (u) |) m (x) T_{j} (u) ϕ | \nabla T_{j} (u) | \\ - \int_{Ω} T_{j} (u) | \nabla T_{j} (u) | log (1 + | \nabla T_{j} (u) |) ϕ . \end{array}$ and we can simplify the first term on the left hand side with the second term of the right hand side to have that $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u \nabla ϕ + ω_{j} ⩾ \int_{Ω} f ϕ - \int_{Ω} T_{j} (u) | \nabla T_{j} (u) | log (1 + | \nabla T_{j} (u) |) ϕ . \end{array}$ Now, letting $j \to \infty$ we get, thanks to the dominated convergence theorem, that $\begin{array}{l} (33) & \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u \nabla ϕ + \int_{Ω} u | \nabla u | log (1 + | \nabla u |) ϕ ⩾ \int_{Ω} f ϕ . \end{array}$ Summarizing (22) and (33) we get $\begin{array}{l} (34) & \int_{Ω} \frac{log (1 + | \nabla u |)}{| \nabla u |} m (x) \nabla u \nabla ϕ + \int_{Ω} u | \nabla u | log (1 + | \nabla u |) ϕ = \int_{Ω} f ϕ, \forall ϕ \in W_{0}^{1, \infty} (Ω) \end{array}$ and then, by density of $W_{0}^{1, \infty} (Ω)$ in $W_{0}^{1, A} (Ω)$ , (2) holds true.
3. Bounded solutions

In this section we will assume a more regular datum and we will prove Corollary 1.2. Consider as test function in (9) $v = G_{k} (u_{n})$ $\begin{array}{l} \frac{1}{n} \int_{Ω} \nabla u_{n} \nabla G_{k} (u_{n}) + \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla G_{k} (u_{n}) \\ + \int_{Ω} u_{n} | \nabla u_{n} | log (1 + | \nabla u_{n} |) | G_{k} (u_{n}) = \int_{Ω} f_{n} G_{k} (u_{n}) . \end{array}$ Drop by positivity the first term and the lower order term to get $\begin{array}{l} \int_{Ω} \frac{log (1 + | \nabla u_{n} |)}{| \nabla u_{n} |} m (x) \nabla u_{n} \nabla G_{k} (u_{n}) ⩽ \int_{Ω} f_{n} G_{k} (u_{n}) . \end{array}$ Hence, using (3), $\begin{array}{l} (35) & \int_{Ω} log (1 + | \nabla G_{k} (u_{n}) |) | \nabla G_{k} (u_{n}) | ⩽ \frac{1}{α} \int_{Ω} f G_{k} (u_{n}) \end{array}$ and (35) is the starting point of the $L^{\infty} (Ω)$ estimate in [4].

Footnotes

Appendix A.

Here we recall some results about Orlicz spaces. For $s ⩾ 0$ we define $\begin{array}{l} A (s) = s log (1 + s) \end{array}$ and we introduce the Lebesgue–Orlicz space $\begin{array}{l} (36) & L^{A} (Ω) = {u : Ω \to R measurable : \exists L > 0, \int_{Ω} A (\frac{| u |}{L}) < \infty} \end{array}$ that is a Banach space with the norm $\begin{array}{l} ‖ u ‖_{L^{A} (Ω)} = inf {λ > 0 : \int_{Ω} A (\frac{| u |}{λ}) < 1} . \end{array}$ The convex conjugate function of A (Fenchel transform) is $\begin{array}{l} \tilde{A} (t) = sup_{s ⩾ 0} {s t - A (s)} . \end{array}$ It can be shown that $\begin{array}{l} s t ⩽ A (s) + \tilde{A} (t), \end{array}$ and then, by definition, the following estimate holds true $\forall s, t \in R^{+}$ $\begin{array}{l} s t ⩽ e^{s} - s - 1 + (1 + t) log (1 + t) - t ⩽ e^{s} - 1 + t log (1 + t) . \end{array}$

If $\begin{array}{l} A (s) = \int_{0}^{s} a (t) d t \end{array}$ then $\begin{array}{l} \tilde{A} = \int_{0}^{s} a^{- 1} (t) d t . \end{array}$ Furthermore, we recall that ${(L^{A} (Ω))}^{'} = L^{\tilde{A}} (Ω)$ and in this case $\begin{array}{l} \tilde{A} (s) = e^{s} - 1 . \end{array}$ From definitions we have: $\begin{array}{l} (37) & \int_{Ω} \frac{u}{‖ u ‖_{L^{A}}} \frac{v}{‖ v ‖_{L^{\tilde{A}}}} ⩽ \int_{Ω} A (\frac{u}{‖ u ‖_{L^{A}}}) + \int_{Ω} \tilde{A} (\frac{v}{‖ v ‖_{L^{\tilde{A}}}}) ⩽ 1 + 1 = 2, \end{array}$ hence $\begin{array}{l} (38) & \int_{Ω} u v ⩽ 2 ‖ u ‖_{L^{A}} ‖ v ‖_{L^{\tilde{A}}} . \end{array}$ The following inequality puts in relation the two norms $‖ \cdot ‖_{1}$ and $‖ \cdot ‖_{A}$ : $\begin{array}{l} (39) & ‖ u ‖_{A} ⩽ max {1, {‖ A (u) ‖}_{1}} . \end{array}$ Finally, we define the Orlicz–Sobolev space $W^{1, A} (Ω)$ as the space of the functions that belong to $L^{A} (Ω)$ whose distributional derivatives belong to $L^{A} (Ω)$ , and with $W_{0}^{1, A} (Ω)$ the closure of $C_{0}^{\infty} (Ω)$ with respect to the $W^{1, A} (Ω)$ norm.

Appendix B.

The existence of a minimizer u for the functional I, defined in (6), is a consequence of the Tonelli–Morrey’s theorem (see for example [7]). Here we repeat the proof in [7], with a little variation given by the assumption of only measurability of the function a with respect to the variable x.

Acknowledgements

The author is grateful to Francesco Petitta for his help and for his valuable advice. The author would like to thank Andrea Cianchi for his suggestions on the references of the Theorem .

References

Boccardo, A contribution to the theory of quasilinear elliptic equations and application to the minimization of integral functionals, Milan Journal of Mathematics 99 (2017), 1–15.

Boccardo and

Gallouët, Strongly nonlinear elliptic equations having natural growth terms and

L^{1}

data, Nonlinear Anal. 19 (1992), 573–579. doi:10.1016/0362-546X(92)90022-7.

Boccardo,

Murat and

J.-P.

Puel,

L^{\infty}

estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. 23 (1992), 326–333. doi:10.1137/0523016.

Boccardo and

Orsina, Leray–Lions operators with logarithmic growth, Journal of Mathematical Analysis and Applications. 423 (2015), 608–622. doi:10.1016/j.jmaa.2014.09.065.

Breit,

De Maria and

Passarelli di Napoli, Regularity for non-autonomous functionals with almost linear growth, Manuscripta Math. 136 (2011), 83–114. doi:10.1007/s00229-011-0432-2.

Brezis, Analisi Funzionale, Liguori editore, Napoli, 2006.

Giaquinta,

Modica and

Souček, Cartesian Currents in the Calculus of Variations II, Springer, Berlin Heidelberg, 1998.

J.-P.

Gossez, Nonlinear elliptic boundary-value problems with rapidly (or slowly) increasing coefficients, Transactions of the American Mathematical Society 190 (1974), 163–205. doi:10.1090/S0002-9947-1974-0342854-2.

J.-P.

Gossez, Orlicz–Sobolev spaces and nonlinear elliptic boundary value problems, in: Proc. Spring School on Funct. Spaces,

Kufner, ed., Vol. 1, Teubner Verlag, Leipzig, 1978, pp. 59–94.

10.

Greco,

Iwaniec and

Sbordone, Variational integrals of nearly linear growth, Differential and Integral Equations 10 (1997), 687–716.

11.

Leray and

J.L.

Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty–Browder, Bull. Soc. Math (France) 93 (1965), 97–107. doi:10.24033/bsmf.1617.

12.

Mingione and

Siepe, Full

C^{1, α}

-Regu1arity for minimizers of integral functionals with

L log L

-growth, Z. Anal. Anwend. 18 (1999), 1083–1100. doi:10.4171/ZAA/929.

13.

Passarelli di Napoli, Existence and regularity results for a class of equations with logarithmic growth, Nonlinear Analysis 125 (2015), 290–309. doi:10.1016/j.na.2015.05.025.

14.

Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258. doi:10.5802/aif.204.