Entropy and renormalized solutions to the general nonlinear elliptic equations in Musielak–Orlicz spaces

Abstract

In this paper we mainly prove the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the general nonlinear elliptic equations in Musielak-Orlicz spaces. Moreover, we also obtain the equivalence of entropy solutions and renormalized solutions in the present conditions.

Introduction

Suppose that $\Omega$ is a bounded Lipschitz domain of ${\mathbb{R}}^N$ with $N>1$ and $f\in L^1\left(\Omega \right)$. In this paper we study the following nonlinear problem

$$\left\{\begin{array}{c}\hfill -\text{div}A(x,\nabla u)=f\text{in}\Omega ,\hfill \\ \hfill u\left(x\right)=0\text{on}\partial \Omega ,\hfill \end{array}\right.$$

where the function $A(x,\xi ):\Omega \times {\mathbb{R}}^N\to {\mathbb{R}}^N$ satisfies the following conditions:

Condition (A1): $A(x,\xi )$ is a Carathéodory function (i.e., measurable with respect to $x$ and continuous with respect to $\xi$).

Condition (A2): There exists an $N$-function $M:\Omega \times {\mathbb{R}}^N\to \mathbb{R}$ and a constant $c_A\in (0,1]$ such that for all $\xi \in {\mathbb{R}}^N$ and a.e. $x\in \Omega$ we have

$$A(x,\xi )\cdot \xi \ge c_A\left[M(x,\xi )+M^{\ast }(x,A(x,\xi ))\right],$$

where $M^{\ast }$ is the complementary function of the modular function $M$

$$M^{\ast }(x,\eta )≔\underset{\xi \in {\mathbb{R}}^N}{sup}\left[\xi \cdot \eta -M(x,\xi )\right]\text{for any}\eta \in {\mathbb{R}}^N\text{and}\text{a.e.}x\in \Omega .$$

Condition (A3): $\left[A(x,\xi )-A(x,\eta )\right]\cdot (\xi -\eta )>0$ for all $\xi ,\eta$ in ${\mathbb{R}}^N$, $\xi \ne \eta$ and a.e. $x\in \Omega$.

Definition 1.1

A function $M(x,\xi ):\Omega \times {\mathbb{R}}^N\to \mathbb{R}$ is called an $N$-function if it satisfies the following conditions:

1. $M(x,\xi )$ is a Carathéodory function satisfying $M(x,0)=0$, ${inf}_{x\in \Omega }M(x,\xi )>0$ for $\xi \ne 0$ and $M(x,\xi )=M(x,-\xi )$ for a.e. $x\in \Omega$.

2. $M(x,\xi )$ is a convex function with respect to $\xi$.

3. ${lim}_{\left|\xi \right|\to 0}\text{ess}{sup}_{x\in \Omega }\frac{M(x,\xi )}{\left|\xi \right|}=0$.

4. ${lim}_{\left|\xi \right|\to \infty }\text{ess}{inf}_{x\in \Omega }\frac{M(x,\xi )}{\left|\xi \right|}=\infty$.

Unlike other studies, we use the following (M) condition on the modular function $M(x,\xi )$ instead of growth conditions. In this paper, $M(x,\xi )$ is not assumed to satisfy $M\in {\Delta }_2$, nor $M^{\ast }\in {\Delta }_2$, nor any particular growth of $M$ such as $M(x,\xi )\ge c{\left|\xi \right|}^{1+\nu }$. Actually, we say an $N$-function $M:\Omega \times {\mathbb{R}}^N\to \mathbb{R}$ satisfies the ${\Delta }_2$ condition if for a.e. $x\in \Omega$, there exists a nonnegative integrable function $h:\Omega \to \mathbb{R}$, such that

$$M(x,2\xi )\le cM(x,\xi )+h\left(x\right)\text{for a constant}c\ge 0.$$

Condition (M): Suppose for every measurable set $G\subset \Omega$ and every $z\in {\mathbb{R}}^N$ we have

$${\int }_{G}M(x,z)dx<\infty .$$

Moreover, there exists a family ${\left\{Q_j^{\delta }\right\}}_{j=1}^{N_{\delta }}$, where $Q_j^{\delta }$ is an N dimensional closed cube of edge $2\delta$, with

$$\text{int}{Q}_j^{\delta }\cap \text{int}{Q}_i^{\delta }=0̸\text{for}i\ne j\text{and}\Omega \subset \bigcup _{j=1}^{N_{\delta }}{Q}_j^{\delta }.$$

Assume that there exist constants $a,b,c,{\delta }_0>0$ such that for all $\delta <{\delta }_0$, $x\in Q_j^{\delta }$ and all $\xi \in {\mathbb{R}}^N$ we have

$$\frac{M(x,\xi )}{{\left(M_j^{\delta }\left(\xi \right)\right)}^{\ast \ast }}\le c\left(1+{\left|\xi \right|}^{-\frac{a}{log\left(b\delta \right)}}\right),$$

where

$$M_j^{\delta }\left(\xi \right)≔\underset{x\in 2Q_j^{\delta }\bigcap \Omega }{inf}M(x,\xi )$$

and ${\left(M_j^{\delta }\left(\xi \right)\right)}^{\ast \ast }={\left({\left(M_j^{\delta }\left(\xi \right)\right)}^{\ast }\right)}^{\ast }$ is the greatest convex minorant of $M_j^{\delta }\left(\xi \right)$.

Actually, in the isotropic case $M(x,\xi )=M(x,\left|\xi \right|)$, we can replace the condition ($M$) by the following log-Hölder continuity condition

$$\frac{M(x,\xi )}{M(y,\xi )}\le max\left\{{\left|\xi \right|}^{-\frac{a_1}{log|x-y|}},b_1^{-\frac{a_1}{log|x-y|}}\right\}$$

for any $\xi \in {\mathbb{R}}^N$, any $x,y$ with $|x-y|<\frac{1}{2}$ and some $a_1>0,b_1\ge 1$. For example,

$$M(x,\xi )=\sum _{i=1}^j{k}_i\left(x\right)M_i\left(\xi \right)+M_0(x,\left|\xi \right|)\text{for}j\in \mathbb{N},$$

where $M_0$ is log-Hölder continuous, $M_i$ are $N$-functions for $i=1,\dots ,j$ and $k_i>0$ satisfy $\frac{k_i\left(x\right)}{k_i\left(y\right)}\le C_i^{log\frac{1}{|x-y|}}$ with $C_i>0$ for $i=1,\dots ,j$.

In recent decades, elliptic and parabolic problems with nonstandard growth became an increasing interest. They arose in the fields of electro-rheological fluid dynamics, elastic mechanics, image processing and so on. Next, we shall give some examples of modular functions $M$ with nonstandard growth, which satisfy condition ($M$). Let us point out here that if modular function $M(x,\xi )=M\left(\xi \right)$, then it satisfies condition $\left(M\right)$ naturally.

Example 1.1

$$M(x,\xi )={\left|\xi \right|}^{p\left(x\right)},$$

where $p\left(x\right):\Omega \to [p^{-},p^{+}]$ is log-Hölder continuous for $p^{-}≔{inf}_{x\in \Omega }p\left(x\right)>1$ and $p^{+}≔{sup}_{x\in \Omega }p\left(x\right)<\infty$.

Example 1.2

$$M(x,\xi )=\sum _{i=1}^N{\left|\xi \right|}^{p_i\left(x\right)},$$

where $p_i\left(x\right):\Omega \to [p_i^{-},p_i^{+}]$ are log-Hölder continuous for $p_i^{-}≔{inf}_{x\in \Omega }{p}_i\left(x\right)>1$ and $p_i^{+}≔{sup}_{x\in \Omega }{p}_i\left(x\right)<\infty$.

Example 1.3

$$M(x,\xi )=a\left(x\right)(exp\left(\left|\xi \right|\right)-1-\left|\xi \right|),$$

where $0<\lambda \le a\left(x\right)\in L^{\infty }\left(\Omega \right)\cap C\left(\Omega \right)$.

Example 1.4

$$M\left(\xi \right)=M_1\left({\xi }_1\right)+M_2\left({\xi }_2\right)+M_3({\xi }_1-{\xi }_2),$$

where $\xi =({\xi }_1,{\xi }_2)\in {\mathbb{R}}^2$, $M_1,M_2,M_3:\mathbb{R}\to [0,\infty )$ are $1$-dimensional $N$-functions satisfying

$$M_{-}=min\{M_1,M_2,M_3\}\le M_1,M_2,M_3\le max\{M_1,M_2,M_3\}=M_{+},$$

where $M_{-}\left(t\right)={\left|t\right|}^p$, $M_{+}\left(t\right)={\left|t\right|}^p{log}^{\alpha }(\left|t\right|+1)$ with $p\ge 1$, $\alpha >0$ and $M_i$ is incomparable with $M_j+M_k$ for any distinct $1\le i,j,k\le 3$. It is an example of fully anisotropic modular function. For more details about this function, we refer the reader to Example 1 in [1].

Remark 1.2

If $M$ is an $N$-function and $A$ satisfies the conditions $\left(A1\right)$–$\left(A3\right)$, then from the elementary Fenchel–Young inequality

$$\left|\xi \cdot \eta \right|\le M(x,\xi )+M^{\ast }(x,\eta )\text{for all}\xi ,\eta \in {\mathbb{R}}^N\text{and}a.e.x\in \Omega ,$$

we have

$$c_A\left[M(x,\xi )+M^{\ast }(x,A(x,\xi ))\right]\le A(x,\xi )\cdot \xi \le M(x,\xi )+M^{\ast }(x,A(x,\xi )).$$

The notion of renormalized solutions was early introduced by DiPerna and Lions [2] for the study of Boltzmann equation. It was then adapted to the study of some nonlinear elliptic problems in [3], [4], [5], [6] and parabolic problems in [7], [8], [9], [10], [11]. At the same time the notion of entropy solutions has been proposed by Bénilan et al. in [12] for the nonlinear elliptic problems. There have been wide research activities [13], [14], [15], [16], [17] on the entropy solutions. It is worth mentioning that in [18], Droniou and Prignet proved the equivalence of entropy solutions and renormalized solutions for parabolic equations with polynomial growth. Furthermore, Zhang and Zhou [19] obtained the equivalence of entropy solutions and renormalized solutions for the nonlinear parabolic equation with variable exponents and $L^1$ data. For more equivalence results of entropy solutions and renormalized solutions, we refer to [20], [21], [22], [23] for details.

Motivated by certain questions in analysis, Musielak–Orlicz spaces have been studied systematically and developed in many fields of mathematics by many authors. Some approximation theorems in Musielak–Orlicz spaces were investigated in [24], [25], [26]. The existence of weak solutions in Musielak–Orlicz spaces was given in [27], [28], [29], [30], [31]. Recently, Gwiazda, Skrzypczak and Zatorska–Goldstein [32] showed the existence of renormalized solutions for problem (1.1) in Musielak–Orlicz spaces. Subsequently, the well-posedness of the renormalized solutions for the parabolic equations in the nonhomogeneous and non-reflexive Musielak–Orlicz spaces was obtained by Chlebicka, Gwiazda and Zatorska–Goldstein in [33]. Moreover, they [34] also studied the renormalized solutions to the parabolic equations in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev’s phenomenon. Furthermore, Denkowska, Gwiazda and Kalita [35] proved the existence and uniqueness of renormalized solutions for the elliptic inclusions with a multivalued leading term in the setting of nonreflexive and nonseparable Musielak–Orlicz spaces. Actually, it is a generalization of the result of [32] to the situation when the single valued mapping $A$ becomes a multivalued map. For more results about the existence of the renormalized solution in Musielak–Orlicz spaces, we would like to mention the papers [36], [37], [38]. In addition, there have been many articles about the existence of entropy solutions in Musielak–Orlicz spaces. In [39], Al-Hawmi et al. obtained the existence and uniqueness of entropy solutions for some quasilinear degenerate elliptic unilateral problems in the framework of Musielak–Orlicz–Sobolev spaces, where the conjugate function of the $N$-function $M(x,\xi )$ satisfies ${\Delta }_2$-condition. Existence of entropy solutions for the nonlinear elliptic problems with lower order term in Musielak–Orlicz spaces has been investigated by the authors in [40], [41]. An existence result of entropy solutions for a class of strongly nonlinear parabolic problems in Musielak–Orlicz–Sobolev spaces was obtained by Elemine Vall et al. [42]. In particular, an equivalence result of two kinds very weak solutions to measure data problems in the reflexive Musielak–Orlicz spaces was established in [43] by Chlebicka. Inspired by the above papers, we want to prove the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to problem (1.1) under the similar assumptions as those in [32]. Moreover, we also obtain the equivalence of entropy solutions and renormalized solutions in the present conditions. Meanwhile, a comparison result is also discussed.

Let $T_k$ denote the truncation function at height $k\ge 0$:

$$T_k\left(r\right)=min\left\{k,max\left\{r,-k\right\}\right\}=\left\{\begin{array}{cc}\hfill r\hfill & \hfill \text{if}\left|r\right|\le k,\hfill \\ \hfill k\frac{r}{\left|r\right|}\hfill & \hfill \text{if}\left|r\right|\ge k.\hfill \end{array}\right.$$

Define

$$V_0^M≔\left\{u\in W_0^{1,1}\left(\Omega \right):\nabla u\in L_M(\Omega ;{\mathbb{R}}^N)\right\},$$

where $L_M\left(\Omega \right)$ is the Musielak–Orlicz space defined by Definition 2.1. Note that as a consequence of Lemma 2.1 of [12], for every measurable function $u$ on $\Omega$ such that $T_k\left(u\right)\in V_0^M$ for every $k>0$, there exists a unique measurable function $Z_u:\Omega \to {\mathbb{R}}^N$ such that

$$\nabla T_k\left(u\right)={\chi }_{\{\left|u\right|<k\}}{Z}_u\text{ for almost every}x\in \Omega \text{and for every}k>0,$$

where ${\chi }_E$ denotes the characteristic function of a measurable set $E$. We denote $Z_u=\nabla u$, which is called the generalized gradient of $u$. The notion of the generalized gradient allows us to give the following definitions of entropy solutions and renormalized solutions for problem (1.1).

Definition 1.3

A function $u$ is an entropy solution to problem (1.1) if $u$ satisfies the following two conditions:

($E1$) $u:\Omega \to \mathbb{R}$ is a measurable function satisfying

$$T_k\left(u\right)\in V_0^M\text{for each}k>0andA(x,\nabla T_k\left(u\right))\in L_{M^{\ast }}(\Omega ;{\mathbb{R}}^N).$$

($E2$) For any $\varphi \in V_0^M\cap L^{\infty }\left(\Omega \right)$ inequality

$${\int }_{\Omega }A(x,\nabla u)\cdot \nabla T_k(u-\varphi )dx\le {\int }_{\Omega }f{T}_k(u-\varphi )dx$$

holds.

Definition 1.4

A function $u$ is a renormalized solution to problem (1.1) if it satisfies the following two conditions:

($R1$) $u:\Omega \to \mathbb{R}$ is a measurable function satisfying

$$T_k\left(u\right)\in V_0^M\text{for each}k>0\text{,}A(x,\nabla T_k\left(u\right))\in L_{M^{\ast }}(\Omega ;{\mathbb{R}}^N)$$

and

$${\int }_{\{l<\left|u\right|<l+1\}}A(x,\nabla u)\cdot \nabla udx\to 0asl\to \infty .$$

($R2$) For any $h\in C_0^1\left(\mathbb{R}\right)$ and $\varphi \in V_0^M\cap L^{\infty }\left(\Omega \right)$ equality

$${\int }_{\Omega }A(x,\nabla u)\cdot \nabla \left(h\left(u\right)\varphi \right)dx={\int }_{\Omega }fh\left(u\right)\varphi dx$$

holds.

Now let us state the main results of this work. The first two theorems are about the existence and uniqueness of renormalized and entropy solutions. The third one is about the comparison principle.

Theorem 1.5

Assume that $f\in L^1\left(\Omega \right)$, $M$ is an $N$-function satisfying the condition $\left(M\right)$ and $A$ satisfies the conditions $\left(A1\right)$–$\left(A3\right)$. Then there exists a unique renormalized solution for problem (1.1).

Theorem 1.6

Assume that $f\in L^1\left(\Omega \right)$, $M$ is an $N$-function satisfying the condition $\left(M\right)$ and $A$ satisfies the conditions $\left(A1\right)$–$\left(A3\right)$. Then the renormalized solution for problem (1.1) is also an entropy solution for problem (1.1). Moreover, the entropy solution is unique.

Remark 1.7

As a consequence, the renormalized solution for problem (1.1) is equivalent to the entropy solution for problem (1.1).

Theorem 1.8

Suppose that $u$ is the entropy solution of problem (1.1). If $f\ge 0$, then we have $u\ge 0$.

The rest of the paper is organized as follows. In Section 2, we state some basic results that will be used later. Section 3 is devoted to the proofs of main results. In the following statement $C$ stands for a constant, which may vary even within the same inequality.