Existence of solutions to a class of elliptic equations with nonstandard growth condition and zero order term

Abstract

The existence of bounded weak solutions, to a class of nonlinear elliptic equations with variable exponents, is investigated in this article. A uniform a priori L^{\mathrm{\infty }} estimate is obtained by the De Giorgi iterative technique. Thanks to the weak convergence method and Minty's trick, the existence result is proved through limit process.

Keywords

Elliptic equations / variable exponents / De Giorgi iteration / Minty's trick

1 Introduction

Let \mathrm{\Omega } be a bounded domain of {\mathbb{R}}^N with a smooth boundary \mathrm{\partial }\mathrm{\Omega }. We consider the following elliptic equation:

(1)\begin{array}{l}\{\begin{array}{l}-\text{div}A(x,u,\mathrm{\nabla }u)+g(u)=-\text{div}\overrightarrow{F}(x),x\in \mathrm{\Omega },\\ u(x)=0,x\in \mathrm{\partial }\mathrm{\Omega }.\end{array}\end{array}

The hypotheses on (1) are given as follows.

\mathbf{(}\mathbf{H}\mathbf{1}\mathbf{)}\mathbf{:} p(x) is a continuous function on \overline{\mathrm{\Omega }}; and

\begin{array}{l}{p}^{+}:=\underset{\overline{\mathrm{\Omega }}}{max}p(x),p^{-}:=\underset{\overline{\mathrm{\Omega }}}{min}p(x),\end{array}

where . Moreover, p(x) satisfies the log-Hölder continuity condition: there exists a constant C>0, such that

(2)|p(x)-p(y)|⩽\frac{-C}{\log |x-y|},

for \mathrm{\forall }x,y\in \mathrm{\Omega }, |x-y|⩽\frac{1}{2}.

\mathbf{(}\mathbf{H}\mathbf{2}\mathbf{)}\mathbf{:} A(x,s,\xi ):\mathrm{\Omega }\times \mathbb{R}\times {\mathbb{R}}^N\to {\mathbb{R}}^N is a vector function which satisfying the Carathéodory condition, i.e., for all (s,\xi )\in \mathbb{R}\times {\mathbb{R}}^N, A(\cdot ,s,\xi ) is measurable on \mathrm{\Omega }; for almost all x\in \mathrm{\Omega }, A(x,\cdot ,\cdot ) is continuous on \mathbb{R}\times {\mathbb{R}}^N. Furthermore, A has the following structure conditions:

(3)\begin{array}{l}[A(x,s,\xi )-A(x,s,{\xi }^{\mathrm{\prime }})]\cdot (\xi -{\xi }^{\mathrm{\prime }})⩾0;\end{array}

(4)\begin{array}{l}A(x,s,\xi )\cdot \xi ⩾\alpha |\xi {|}^{p(x)};\end{array}

(5)\begin{array}{l}|A(x,s,\xi )|⩽b(|s|)|\xi {|}^{p(x)-1},\end{array}

where \alpha >0; \xi ,{\xi }^{\mathrm{\prime }}\in {\mathbb{R}}^N. The function b:[0,+\mathrm{\infty })\to (0,+\mathrm{\infty }) is assumed to be continuous.

\mathbf{(}\mathbf{H}\mathbf{3}\mathbf{)}\mathbf{:} \overrightarrow{F}(x) is a vector field; and |\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}\in L^q(\mathrm{\Omega }) with q>\frac{N}{p^{-}}.

\mathbf{(}\mathbf{H}\mathbf{4}\mathbf{)}\mathbf{:} g(s) is a nondecreasing and continuous function on \mathbb{R}, and it satisfies

(6)\begin{array}{l}g(s)s⩾0.\end{array}

The log-Hölder continuity condition (2), stated in (H1), ensures that C_0^{\mathrm{\infty }}(\mathrm{\Omega }) is dense in W_0^{1,p(x)}(\mathrm{\Omega }) (see the details in [11]).

In order to describe the nonlinear diffusion problem more refined, elliptic and parabolic equations with variable exponents are powerful tools. For more applications in physics, the variable exponent problem is related to the electro-rheological fluids. As pointed out in [15], when subjected to an external electric field, this kind of fluids can give a well description of the ability of a conductor to withstand drastic change of the electric current.

There are numerous papers for this kind of problem like Eq. (1). When the zero order term g=0 and the right-hand side is only L^1 integrable in Eq. (1), with the use of the truncation method and monotone operator theory, Bendahmane and Wittbold proved the existence and uniqueness of a class of elliptic p(x)-Laplace equation in [2]. For the result concerning parabolic equations, we refer to [3, 17] and the references therein.

In the case of p(x) being a constant and A being a strictly monotone operator, Boccardo and Croce investigated Problem (1) systematically in the monograph [5]. In [4], when g=0, Betta, Guibé and Mercaldo studied the existence and stability of the Neumann boundary problem with a non-coercive term -\text{div}(c(x)|u{|}^{p-2}u) and an L^1 source. A(u)=-\sum _{i=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}(|u_{x_i}{|}^{p_i-2}{u}_{x_i}) is assumed in [6], the authors considered the case of anisotropic elliptic equation and the corresponding variational problem. The L^{\mathrm{\infty }} regularity of this nonstandard growth problem is proved by virtue of Sobolev embedding with anisotropy and the iterative technique. For the maximal norm estimate to the anisotropic parabolic equations, one can refer to [14]. In the framework of Orlicz spaces, a class of elliptic equations, with a principal diffusion term

-\text{div}\left[\frac{\log (1+|\mathrm{\nabla }u|)}{|\mathrm{\nabla }u|}\mathrm{\nabla }u\right],

is investigated in [9]. With the help of Dunford-Pettis Theorem, the authors obtained the weak convergence in L^1 for the gradient solution sequence, as well as the existence of the weak solutions. [10] proved the existence result to a class of parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function.

When the operator A is strictly monotone, in order to obtain the existence of solution, as in [7], we usually seek the strong convergence or almost everywhere convergence of the gradient sequence \{\mathrm{\nabla }{u}_k{\}}_k. However, the operator A in our Problem (1) is only monotone. Compared with the strict monotonicity of operator A, it is not easy to get the strong convergence or almost everywhere convergence of gradient sequence. In essence, it is more complicated to deal with the weak convergence of the nonlinear term.

Theorem 1　 If Problem (1) satisfy conditions (H1), (H2), (H3) and (H4), then there exists a bounded weak solution u\in W_0^{1,p(x)}(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega }), i.e., for arbitrary \varphi \in W_0^{1,p(x)}(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega }), there holds

{\int }_{\mathrm{\Omega }}A(x,u,\mathrm{\nabla }u)\cdot \mathrm{\nabla }\varphi \mathrm{d}x+{\int }_{\mathrm{\Omega }}g(u)\varphi \mathrm{d}x={\int }_{\mathrm{\Omega }}\overrightarrow{F}(x)\cdot \mathrm{\nabla }\varphi \mathrm{d}x.

2 Proof of Theorem 1

2.1 Regularized problem

Denote

T_k(s)=min\{k,max(s,-k)\},k\in \mathbb{N};F_k(x)={\displaystyle \frac{F(x)}{1+{\displaystyle \frac{1}{k}}|F(x)|}}.

Inspired by [4, 9], we construct a regularized equation corresponding to Problem (1):

(7)\begin{array}{l}\{\begin{array}{l}-{\displaystyle \frac{1}{k}}\text{div}(|\mathrm{\nabla }{u}_k{|}^{p(x)-2}\mathrm{\nabla }{u}_k)-\text{div}A(x,T_k(u_k),\mathrm{\nabla }{u}_k)+T_k(g(u_k))=-\text{div}\overrightarrow{F_k}(x),x\in \mathrm{\Omega },\\ u_k(x)=0,x\in \mathrm{\partial }\mathrm{\Omega }.\end{array}\end{array}

Denote

f_k=k[\text{div}A(x,T_k(u_k),\mathrm{\nabla }{u}_k)-T_k(g(u_k))-\text{div}\overrightarrow{F_k}(x)],\frac{1}{p(x)}+\frac{1}{p^{\mathrm{\prime }}(x)}=1.

Then f_k lies in W^{-1,p^{\mathrm{\prime }}(x)}(\mathrm{\Omega }), the duality space of W_0^{1,p(x)}(\mathrm{\Omega }).

Based upon the derivation of Section 3.1 and Remark 3.2 in [2], thanks to the monotone operator theory in [16], for every fixed k\in \mathbb{N}, there exists a unique weak solution u_k\in W_0^{1,p(x)}(\mathrm{\Omega }) to Eq. (7). Moreover, [6] shows that for every k, u_k is in L^{\mathrm{\infty }}(\mathrm{\Omega }) (it is not uniformly bounded in L^{\mathrm{\infty }} [7]).

2.2 The result for the case: |\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}\in L^1(\mathrm{\Omega })

When the right-hand side term \overrightarrow{F} is endowed with the integrability |\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}\in L^1(\mathrm{\Omega }), we will prove the necessary a priori estimates for the solution sequence \{u_k\}.

Taking u_k as a test function in Problem (7), from (4), (6) and Young inequality ab⩽\epsilon a^{p(x)}+{\epsilon }^{\frac{-p^{\mathrm{\prime }}(x)}{p(x)}}{b}^{p^{\mathrm{\prime }}(x)}, we obtain that

\begin{array}{rl}& {\int }_{\mathrm{\Omega }}\frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)}\mathrm{d}x+\alpha {\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{u}_k{|}^{p(x)}\mathrm{d}x\\ & ⩽{\int }_{\mathrm{\Omega }}\frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)}\mathrm{d}x+{\int }_{\mathrm{\Omega }}A(x,T_k(u_k),\mathrm{\nabla }{u}_k)\cdot \mathrm{\nabla }{u}_k\mathrm{d}x+{\int }_{\mathrm{\Omega }}{T}_k(g(u_k))u_k\mathrm{d}x\\ & ={\int }_{\mathrm{\Omega }}\overrightarrow{F_k}(x)\cdot \mathrm{\nabla }{u}_k\mathrm{d}x⩽\frac{\alpha }{2}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{u}_k{|}^{p(x)}\mathrm{d}x+(\frac{2}{\alpha }+1{)}^{\frac{1}{p^{-}-1}}{\int }_{\mathrm{\Omega }}|\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}\mathrm{d}x.\end{array}

Then there holds

(8){\int }_{\mathrm{\Omega }}\frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)}dx+\frac{\alpha }{2}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{u}_k{|}^{p(x)}dx⩽{\left(\frac{2}{\alpha }+1\right)}^{\frac{1}{p^{-}-1}}\Vert |\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}{\Vert }_{1,\mathrm{\Omega }}:=\tilde{C}.

This inequality indicates that \{\frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)}{\}}_k is uniformly bounded in L^1(\mathrm{\Omega }). Simultaneously, \{u_k{\}}_k is uniformly bounded in W_0^{1,p(x)}(\mathrm{\Omega }). With the help of (4), we also get

(9){\int }_{\mathrm{\Omega }}{T}_k(g(u_k))u_k\mathrm{d}x⩽2\tilde{C}.

The previous estimates imply that there exists a subsequence of \{u_k{\}}_k, still denoted by \{u_k{\}}_k itself, and a u\in W_0^{1,p(x)}(\mathrm{\Omega }), such that for k\to \mathrm{\infty },

(10)\begin{array}{rl}& u_k\rightharpoonup u,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{W}_0^{1,p(x)}(\mathrm{\Omega });\\ & \mathrm{\nabla }{u}_k\rightharpoonup \mathrm{\nabla }u,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{L}^{p(x)}(\mathrm{\Omega };{\mathbb{R}}^N).\end{array}

It follows from the Compactness Embedding Theorem in [11] that

(11)\begin{array}{l}{u}_k\to u,\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{L}^{p(x)}(\mathrm{\Omega }),\mathrm{a}.\mathrm{e}.\mathrm{i}\mathrm{n}\mathrm{\Omega }.\end{array}

We deduce from (11) and the continuity of g that

(12)\begin{array}{l}{T}_k(g(u_k))\to g(u),\mathrm{a}.\mathrm{e}.\mathrm{i}\mathrm{n}\mathrm{\Omega },\end{array}

as k\to \mathrm{\infty }.

Denoted |U| by the Lebesgue measure of the measurable set U. For every \epsilon >0, and for arbitrary measurable subset E\subset \mathrm{\Omega } and \hat{k}\in {\mathbb{R}}^{+}, there holds

(13)\begin{array}{rl}& {\int }_E|T_k(g(u_k))|\mathrm{d}x\\ & ={\int }_{E\cap \{x\in \mathrm{\Omega }:|u_k(x)|⩽\hat{k}\}}|T_k(g(u_k))|\mathrm{d}x+{\int }_{E\cap \{x\in \mathrm{\Omega }:|u_k(x)|>\hat{k}\}}|T_k(g(u_k))|\mathrm{d}x\\ & ⩽{\int }_{E\cap \{x\in \mathrm{\Omega }:|u_k(x)|⩽\hat{k}\}}|g(u_k)|\mathrm{d}x+\frac{1}{\hat{k}}{\int }_{E\cap \{x\in \mathrm{\Omega }:|u_k(x)|>\hat{k}\}}|T_k(g(u_k))||u_k|\mathrm{d}x\\ & ⩽[g(\hat{k})-g(-\hat{k})]|E|+\frac{1}{\hat{k}}{\int }_{\mathrm{\Omega }}{T}_k(g(u_k))u_k\mathrm{d}x.\end{array}

Here, the first term on the right-hand side of (13) is obtained in the following manner: in view of the monotonicity of g and (6),

Noting (9), by choosing \hat{k}=k_0 sufficiently large, one has

As soon as \hat{k}=k_0 is fixed in (13), then by selecting \delta =\frac{\epsilon }{2[g(k_0)-g(-k_0)]+1}, when , we see that

(14)

Combining (12) and (14), from the Vitali Theorem [5] we find that

(15)\begin{array}{l}{T}_k(g(u_k))\to g(u),\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{L}^1(\mathrm{\Omega }).\end{array}

2.3 Uniform L^{\mathrm{\infty }}-estimate

Under the assumption (H3), when |\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}\in L^q(\mathrm{\Omega }), the uniform L^{\mathrm{\infty }} estimate of u_k will be proved.

Denote A_j=\{x\in \mathrm{\Omega }:|u_k(x)|>j\}, G_j(s)=(|s|-j{)}_{+}sign(s). Since for every j, G_j(u_k)\in W_0^{1,p(x)}(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega }), G_j(u_k) is able to be employed as a test function to Problem (7). With the use of (4), (6) and Young inequality, we observe that

\begin{array}{rl}\alpha {\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{G}_j(u_k){|}^{p(x)}\mathrm{d}x& ⩽{\int }_{\mathrm{\Omega }}A(x,T_k(u_k),\mathrm{\nabla }{u}_k)\cdot \mathrm{\nabla }{G}_j(u_k)\mathrm{d}x\\ & ⩽{\int }_{\mathrm{\Omega }}\overrightarrow{F_k}(x)\cdot \mathrm{\nabla }{G}_j(u_k)\mathrm{d}x\\ & ⩽\frac{\alpha }{2}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{G}_j(u_k){|}^{p(x)}\mathrm{d}x+(\frac{2}{\alpha }+1{)}^{\frac{1}{p^{-}-1}}{\int }_{A_j}|\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}\mathrm{d}x.\end{array}

Therefore,

{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{G}_j(u_k){|}^{p(x)}\mathrm{d}x⩽\frac{2}{\alpha }{\left(\frac{2}{\alpha }+1\right)}^{\frac{1}{p^{-}-1}}{\int }_{A_j}|\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}\mathrm{d}x.

Moreover,

\begin{array}{rl}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{G}_j(u_k){|}^{p^{-}}\mathrm{d}x& ⩽\frac{2}{\alpha }{\left(\frac{2}{\alpha }+1\right)}^{\frac{1}{p^{-}-1}}{\int }_{A_j}|\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}\mathrm{d}x+|A_j|\\ & ⩽\frac{2}{\alpha }{\left(\frac{2}{\alpha }+1\right)}^{\frac{1}{p^{-}-1}}\Vert |\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}{\Vert }_{q,\mathrm{\Omega }}|A_j{|}^{\frac{1}{q^{\mathrm{\prime }}}}+|A_j|.\end{array}

Denoted {\beta }_{N,p^{-}} by the Sobolev constant, (p^{-}{)}^{\ast }=\frac{N{p}^{-}}{N-p^{-}}. By means of Sobolev inequality in [1], we have

{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{G}_j(u_k){|}^{p^{-}}\mathrm{d}x⩾\frac{1}{{\beta }_{N,p^{-}}^{p^{-}}}{\left({\int }_{\mathrm{\Omega }}|G_j(u_k){|}^{(p^{-}{)}^{\ast }}\mathrm{d}x\right)}^{\frac{p^{-}}{(p^{-}{)}^{\ast }}}.

Considering the previous two inequalities, one has

{\int }_{\mathrm{\Omega }}|G_j(u_k){|}^{(p^{-}{)}^{\ast }}\mathrm{d}x⩽C_1|A_j{|}^{\frac{1}{q^{\mathrm{\prime }}}\frac{(p^{-}{)}^{\ast }}{p^{-}}}+C_2|A_j{|}^{\frac{(p^{-}{)}^{\ast }}{p^{-}}},

with C_1=2^{\frac{(p^{-}{)}^{\ast }}{p^{-}}}{\beta }_{N,p^{-}}^{(p^{-}{)}^{\ast }}[\frac{2}{\alpha }(\frac{2}{\alpha }+1{)}^{\frac{1}{p^{–}1}}{]}^{\frac{(p^{-}{)}^{\ast }}{p^{-}}}\Vert |\overrightarrow{F}(x){|}^{p^{\mathrm{\prime }}(x)}{\Vert }_{q,\mathrm{\Omega }}^{\frac{(p^{-}{)}^{\ast }}{p^{-}}}, C_2=2^{\frac{(p^{-}{)}^{\ast }}{p^{-}}}{\beta }_{N,p^{-}}^{(p^{-}{)}^{\ast }}.

Let h>j. Then |G_j(u_k)|⩾h-j on the set A_h, we thus arrive at

{\int }_{\mathrm{\Omega }}|G_j(u_k){|}^{(p^{-}{)}^{\ast }}\mathrm{d}x⩾(h-j{)}^{(p^{-}{)}^{\ast }}|A_h|.

In conclusion,

|A_h|⩽\frac{C}{(h-j{)}^{(p^{-}{)}^{\ast }}}\left[|A_j{|}^{\frac{1}{q^{\mathrm{\prime }}}\frac{(p^{-}{)}^{\ast }}{p^{-}}}+|A_j{|}^{\frac{(p^{-}{)}^{\ast }}{p^{-}}}\right],

where C=C_1+C_2.

Recalling (H3), q>\frac{N}{p^{-}}\iff \frac{1}{q^{\mathrm{\prime }}}\frac{(p^{-}{)}^{\ast }}{p^{-}}>1; and it is obvious that \frac{(p^{-}{)}^{\ast }}{p^{-}}>1, by virtue of the modified De Giorgi iterative lemma in [13], we actually prove the maximal norm estimate of u_k:

(16)\begin{array}{l}\underset{k}{sup}\Vert u_k{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}⩽C,\end{array}

where C is independent of k. Furthermore, as k\to \mathrm{\infty },

(17)\begin{array}{l}{u}_k\stackrel{\ast }{\rightharpoonup }u,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\ast \mathrm{i}\mathrm{n}{L}^{\mathrm{\infty }}(\mathrm{\Omega }).\end{array}

From (5) and the uniform L^{\mathrm{\infty }} boundedness (16) of u_k, we can see that

\begin{array}{l}|A(x,T_k(u_k),\mathrm{\nabla }{u}_k)|⩽\underset{r\in [0,\underset{k}{sup}\Vert u_k{\Vert }_{L^{\mathrm{\infty }}}]}{max}b(r)|\mathrm{\nabla }{u}_k{|}^{p(x)-1}.\end{array}

Moreover, according to (8) and the Nemyckii operator theory with variable exponent in [12], we deduce that A(x,T_k(u_k),\mathrm{\nabla }{u}_k) is uniformly bounded in L^{p^{\mathrm{\prime }}(x)}(\mathrm{\Omega };{\mathbb{R}}^N); and

(18)\begin{array}{l}A(x,T_k(u_k),\mathrm{\nabla }{u}_k)\rightharpoonup \sigma ,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{L}^{p^{\mathrm{\prime }}(x)}(\mathrm{\Omega };{\mathbb{R}}^N).\end{array}

2.4 Minty’s trick

Taking advantage of u_k-u as a test function in Problem (7), it results

\begin{array}{rl}& {\int }_{\mathrm{\Omega }}\frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)}\mathrm{d}x-{\int }_{\mathrm{\Omega }}\frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)-2}\mathrm{\nabla }{u}_k\cdot \mathrm{\nabla }u\mathrm{d}x\\ & +{\int }_{\mathrm{\Omega }}A(x,T_k(u_k),\mathrm{\nabla }{u}_k)\cdot \mathrm{\nabla }(u_k-u)\mathrm{d}x\\ & +{\int }_{\mathrm{\Omega }}{T}_k(g(u_k))(u_k-u)\mathrm{d}x={\int }_{\mathrm{\Omega }}\overrightarrow{F_k}(x)\cdot \mathrm{\nabla }(u_k-u)\mathrm{d}x.\end{array}

Taking the lim sup on the previous equality, it follows from (10), (15), (17) that both the limits of the forth and the fifth terms are 0. Besides, observe that \frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)-2}\mathrm{\nabla }{u}_k converges weakly to 0 in L^{p^{\mathrm{\prime }}(x)}(\mathrm{\Omega };{\mathbb{R}}^N). Indeed, for arbitrary \phi \in L^{p(x)}(\mathrm{\Omega };{\mathbb{R}}^N), note the definition of weak convergence, p(x)-Hölder inequality and the relationship between norm and modular in the variable exponent space, by (8) we have

\begin{array}{rl}& \left|{\int }_{\mathrm{\Omega }}\frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)-2}\mathrm{\nabla }{u}_k\cdot \phi \mathrm{d}x\right|⩽{\int }_{\mathrm{\Omega }}{\left(\frac{1}{k}\right)}^{\frac{1}{p^{\mathrm{\prime }}(x)}}|\mathrm{\nabla }{u}_k{|}^{p(x)-1}{\left(\frac{1}{k}\right)}^{\frac{1}{p(x)}}|\phi |\mathrm{d}x\\ & ⩽2{\Vert {\left(\frac{1}{k}\right)}^{\frac{1}{p^{\mathrm{\prime }}(x)}}|\mathrm{\nabla }{u}_k{|}^{p(x)-1}\Vert }_{L^{p^{\mathrm{\prime }}(x)}(\mathrm{\Omega })}{\Vert {\left(\frac{1}{k}\right)}^{\frac{1}{p(x)}}|\phi |\Vert }_{L^{p(x)}(\mathrm{\Omega })}\\ & ⩽2{\left({\int }_{\mathrm{\Omega }}\frac{1}{k}|\mathrm{\nabla }{u}_k{|}^{p(x)}\mathrm{d}x+1\right)}^{\frac{1}{(p^{\mathrm{\prime }}{)}^{-}}}{\left(\frac{1}{k}\right)}^{\frac{1}{p^{+}}}\Vert \phi {\Vert }_{L^{p(x)}(\mathrm{\Omega })}\\ & ⩽\frac{2}{k^{\frac{1}{p^{+}}}}(\tilde{C}+1{)}^{\frac{1}{(p^{\mathrm{\prime }}{)}^{-}}}\Vert \phi {\Vert }_{L^{p(x)}(\mathrm{\Omega })}\to 0.\end{array}

Thus the limit of the second integral term is 0. Utilizing (18) we finally get

(19)\underset{k\to \mathrm{\infty }}{lim sup}{\int }_{\mathrm{\Omega }}A(x,T_k(u_k),\mathrm{\nabla }{u}_k)\cdot \mathrm{\nabla }{u}_k\mathrm{d}x⩽{\int }_{\mathrm{\Omega }}\sigma \cdot \mathrm{\nabla }u\mathrm{d}x.

For arbitrary \psi \in (C_0^{\mathrm{\infty }}(\mathrm{\Omega }){)}^N, taking into account (5), (11), L^{\mathrm{\infty }} boundedness (16) of u_k and the Lebesgue Dominated Convergence Theorem, we infer that

(20)\begin{array}{l}A(x,T_k(u_k),\psi )\to A(x,u,\psi ),\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{L}^{p^{\mathrm{\prime }}(x)}(\mathrm{\Omega };{\mathbb{R}}^N),\end{array}

as k\to \mathrm{\infty }.

In view of the monotonicity (3) of A, together with (10), (18), (19), (20), we deduce that

\begin{array}{rl}0⩽& \underset{k\to \mathrm{\infty }}{lim sup}{\int }_{\mathrm{\Omega }}[A(x,T_k(u_k),\mathrm{\nabla }{u}_k)-A(x,T_k(u_k),\psi )]\cdot (\mathrm{\nabla }{u}_k-\psi )\mathrm{d}x\\ ⩽& \underset{k\to \mathrm{\infty }}{lim sup}{\int }_{\mathrm{\Omega }}A(x,T_k(u_k),\mathrm{\nabla }{u}_k)\cdot \mathrm{\nabla }{u}_k\mathrm{d}x-{\int }_{\mathrm{\Omega }}\sigma \cdot \psi \mathrm{d}x-{\int }_{\mathrm{\Omega }}A(x,u,\psi )\cdot (\mathrm{\nabla }u-\psi )\mathrm{d}x\\ ⩽& {\int }_{\mathrm{\Omega }}[\sigma -A(x,u,\psi )]\cdot (\mathrm{\nabla }u-\psi )\mathrm{d}x.\end{array}

Now selecting \psi =\mathrm{\nabla }u-tw with t>0 and w\in (C_0^{\mathrm{\infty }}(\mathrm{\Omega }){)}^N in the previous inequality, then it allows one to get

{\int }_{\mathrm{\Omega }}[\sigma -A(x,u,\mathrm{\nabla }u-tw)]\cdot w\mathrm{d}x⩾0.

Letting t\to 0, it follows that

{\int }_{\mathrm{\Omega }}[\sigma -A(x,u,\mathrm{\nabla }u)]\cdot w\mathrm{d}x⩾0.

Replacing w by -w, we conclude that

{\int }_{\mathrm{\Omega }}[\sigma -A(x,u,\mathrm{\nabla }u)]\cdot w\mathrm{d}x=0,

which gives the required result

(21)\begin{array}{l}\sigma =A(x,u,\mathrm{\nabla }u)\end{array}

by the Minty's trick.

Once (21) is obtained, by using a standard process of passing to the limit, one can prove that u is a bounded weak solution to Problem (1).

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