Weighted Stochastic Field Exponent Sobolev Spaces and Nonlinear Degenerated Elliptic Problem

TL;DR

This paper introduces weighted stochastic field exponent Sobolev spaces, explores their properties, and applies them to analyze stochastic PDEs with stochastic field growth, advancing the mathematical framework for such problems.

Contribution

The paper develops new weighted stochastic field exponent Sobolev spaces and demonstrates their application to stochastic partial differential equations with stochastic growth conditions.

Findings

Defined new weighted stochastic field exponent spaces

Established properties and embeddings of these spaces

Applied the framework to stochastic PDEs with growth conditions

Abstract

In this study, we consider weighted stochastic field exponent function spaces $L_{\vartheta }^{p(.,.)}\left( D\times \Omega \right) $ and $W_{\vartheta }^{k,p(.,.)}\left( D\times \Omega \right) $. Also, we investigate some basic properties and embeddings of these spaces. Finally, we present an application of these spaces to the stochastic partial differential equations with stochastic field growth.

Equations

$$\left\{\begin{array}[]{cc}-\mathop{\mathrm{d}iv}A\left(x,t,u,\nabla u\right)+A_{0}\left(x,t,u,\nabla u\right)=f\left(x,t\right),&\left(x,t\right)\in D\times\Omega\\ u=0,&\left(x,t\right)\in\partial D\times\Omega\end{array}\right.$$

$$\left\{\begin{array}[]{cc}-\mathop{\mathrm{d}iv}\left(\vartheta(x,t)\left|\nabla u\right|^{p(x,t)-2}\nabla u\right)+\vartheta(x,t)g(u)\left|\nabla u\right|^{p(x,t)-1}=f\left(x,t\right),&\left(x,t\right)\in D\times\Omega\\ u=0,&\left(x,t\right)\in\partial D\times\Omega\end{array}\right..$$

$$p^{-}=\underset{(x,t)\in D\times \Omega }{\mathrm{essinf}}p(x,t)\text{, }{p}^{+}=\underset{(x,t)\in D\times \Omega }{\mathrm{esssup}}p(x,t)\text{.}$$

$$(H_1)$$

$$(H_2)$$

$${\rho }_{p(.,.),\vartheta }\left(u\right)=E\left(\underset{D}{\int }{\left|u\left(x,t\right)\right|}^{p\left(x,t\right)}\vartheta \left(x,t\right)dx\right)=\underset{D\times \Omega }{\int }{\left|u\left(x,t\right)\right|}^{p\left(x,t\right)}\vartheta \left(x,t\right)d\lambda ,$$

$${\Vert u\Vert }_{p(.,.),\vartheta }=\inf \left\{\lambda >0:{\rho }_{p(.,.),\vartheta }\left(\frac{u}{\lambda }\right)\le 1\right\}\text{.}$$

$$E\left(\underset{D}{\int }\left|f(x,t)g(x,t)\right|dx\right)\le C{\Vert f\Vert }_{p(.,.),\vartheta }{\Vert g\Vert }_{q(.,.),{\vartheta }^{\ast }}$$

$$E\left(\underset{D}{\int }\left|f(x,t)g(x,t)\right|dx\right)$$

$${\rho }_{p\left(.,.\right),K}\left(u\right)$$

$${\rho }_{q(.,.),K}({\vartheta }^{-\frac{1}{p(.,.)}})=E\left(\underset{K_1}{\int }{\vartheta }^{-\frac{q(x,t)}{p(x,t)}}dx\right)=\underset{K}{\int }{\vartheta }^{-\frac{1}{p(x,t)-1}}d\lambda =B_K<\infty .$$

$$E\left(\underset{D}{\int }{u}_{n}gdx\right)⟶E\left(\underset{D}{\int }ugdx\right)$$

$$E\left(\underset{D}{\int }{\left|\frac{u}{C}\right|}^{p(x,t)}\vartheta (x,t)dx\right)\le \underset{n⟶\infty }{\mathrm{liminf}}E\left(\underset{D}{\int }{\left|\frac{u_n}{C}\right|}^{p(x,t)}\vartheta (x,t)dx\right)\le 1.$$

$$\underset{meas(K)⟶0}{\lim }\underset{D\times \Omega }{\int }{\left|g{\chi }_K\right|}^{q(x,t)}\vartheta (x,t)d\lambda =0$$

$${\Vert g{\chi }_K\Vert }_{q(.,.),{\vartheta }^{\ast }}<\frac{\epsilon }{4C}{\left(1+\frac{1}{p^{-}}-\frac{1}{p^{+}}\right)}^{-1}$$

$$\underset{(x,t)\in L}{\max }\left|u_n-u\right|{\Vert g\Vert }_{q(.,.),{\vartheta }^{\ast }}{\Vert {\chi }_L\Vert }_{p(.,.),\vartheta }\left(1+\frac{1}{p^{-}}-\frac{1}{p^{+}}\right)<\frac{\epsilon }{2}.$$

$$\left|E\left(\underset{D}{\int }{u}_{n}gdx\right)-E\left(\underset{D}{\int }ugdx\right)\right|$$

$$W_{\vartheta }^{k,p(.,.)}\left(D\times \Omega \right)=\left\{u\in L_{\vartheta }^{p(.,.)}\left(D\times \Omega \right):D^{\alpha }u\in L_{\vartheta }^{p(.,.)}\left(D\times \Omega \right),0\le \left|\alpha \right|\le k\right\}$$

$${\Vert u\Vert }_{W_{\vartheta }^{k,p(.,.)}\left(D\times \Omega \right)}=\sum _{0\le |\alpha |\le k}{\Vert D^{\alpha }u\Vert }_{p(.,.),\vartheta },$$

$$W_{\vartheta }^{1,p(.,.)}\left(D\times \Omega \right)=\left\{u\in L_{\vartheta }^{p(.,.)}\left(D\times \Omega \right):\left|\nabla u\right|\in L_{\vartheta }^{p(.,.)}\left(D\times \Omega \right)\right\}$$

$$C\left(D\times \Omega \right)=\left\{u:u\left(.,t\right)\in C_0^{\infty }\left(D\right)\text{ for each }t\in \Omega \right\}$$

$${\Vert u\Vert }_{p(.,.),\vartheta }\le C{\Vert \nabla u\Vert }_{p(.,.),\vartheta }$$

$${\Vert \left|u\right|\Vert }_{W_{0,\vartheta }^{1,p(.,.)}\left(D\times \Omega \right)}={\Vert \nabla u\Vert }_{p(.,.),\vartheta }$$

$$p^{\ast}\left(x,t\right)=\left\{\begin{array}[]{cc}\frac{dp(x,t)}{d-p(x,t)}&\text{if }p\left(x,t\right)<d,\\ \infty&\text{if }p\left(x,t\right)\geq d,\end{array}\right.,$$

$$p_s\left(x,t\right)=\frac{p(x,t)s(x,t)}{s(x,t)+1}<p(x,t),$$

$$p_{s}^{\ast}\left(x,t\right)=\left\{\begin{array}[]{cc}\frac{dp_{s}(x,t)}{d\left(s(x,t)+1\right)-p(x,t)s(x,t)}&\text{if }p_{s}\left(x,t\right)<d,\\ \text{arbitrary}&\text{if }p_{s}\left(x,t\right)\geq d,\end{array}\right.$$

$$E\left(\underset{D}{\int }\vartheta (x,t){\left|u\right|}^{q(x,t)}dx\right)\le \left(1+\frac{1}{p^{-}}-\frac{1}{p^{+}}\right){\Vert \vartheta \Vert }_{\alpha (.,.)}{\Vert {\left|u\right|}^{q(.,.)}\Vert }_{{\alpha }_0(.,.)}<\infty .$$

$$E\left(\underset{D}{\int }{\left|u_n(x,t)\right|}^{r(x,t)}dx\right)={\rho }_{{\alpha }_0(.,.)}({\left|u_n(x,t)\right|}^{q(x,t)})⟶0$$

$${\Vert {\left|u_n\right|}^{q(.,.)}\Vert }_{{\alpha }_0(.,.)}⟶0.$$

$$E\left(\underset{D}{\int }\vartheta (x,t){\left|u_n\right|}^{q(x,t)}dx\right)\le \left(1+\frac{1}{p^{-}}-\frac{1}{p^{+}}\right){\Vert \vartheta \Vert }_{\alpha (.,.)}{\Vert {\left|u_n\right|}^{q(.,.)}\Vert }_{{\alpha }_0(.,.)}⟶0$$

$$W_{\vartheta }^{1,p(.,.)}\left(D\times \Omega \right)\hookrightarrow \hookrightarrow L_{\vartheta }^{r(.,.)}\left(D\times \Omega \right).$$

Full text

1. Introduction

Nonlinear partial differential equations arise in chemical and biological problems, in formulating fundamental laws of nature, in different areas of physics, applied mathematics and engineering (such as solid mechanics, fluid dynamics, acoustics, nonlinear optics, plasma physics, quantum eld theory) and numerous applications. To study these equations is a very difficult task because there are no general methods to solve such equations. Moreover, the existence and the uniqueness of the solutions are fundamental, hard-to-prove questions for any nonlinear equation with given boundary conditions.

In various applications (such as elasticity, non-Newtonian fluids and electrorheological fluids, see [11]), it can be seen the boundary value obstacle problems for elliptic equations. Many of these type equations have been investigated for constant exponents of nonlinearity but it seems to be more realistic to assume the variable exponent.

Harjulehto et. al [6] investigate an overview of applications to differential equations with non-standard growth. Also, Aoyama [1] discussed the some properties of Lebesgue spaces with variable exponent on a probability space. In 2014, Tian et. al [13] introduced stochastic field exponent function spaces $L^{p(.,.)}\left(D\times\Omega\right)$ and $W^{k,p(.,.)}\left(D\times\Omega\right)$. They gave also an application to the stochastic partial differential equations with stochastic field growth in these spaces. Moreover, Lahmi et. al [8] proved the existence of solutions for the nonlinear $p(.)$-degenerate problems involving nonlinear operators. This study is a generalization of [8], [12] and [13]. Stochastic partial differential equations have many applications in finance, such as option pricing etc.

In this paper, we define weighted stochastic field exponent function spaces $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ and $W_{\vartheta}^{k,p(.,.)}\left(D\times\Omega\right)$, and discuss some basic properties of these spaces. Finally, we discuss the existence and uniqueness of the weak solution for the nonlinear degenerated weighted $p(.,.)$ elliptic problem (the stochastic partial differential equations with stochastic field growth)

[TABLE]

where $A\left(x,t,s,\xi\right)$ is a Carathéodory function, which is measurable stochastic fields on $D\times\Omega$ and continuous for $s$ and $\xi$ under some conditions.

It is known that pseudo-monotone operators have been many applications in nonlinear elliptic equations. For example, Browder [2] studied a class of pseudo-monotone operators and applied it to a kind of boundary value problems for nonlinear elliptic equations.

By the theory of pseudo-monotone operators, our aim is to show the compactness techniques and the existence of a least weak solution of (1.1). The special case of the equation of (1.1) is the following equation

[TABLE]

Let $\lambda$ be a product measure on $D\times\Omega$ and $u(x,t)$ be a Lebesgue measurable stochastic field on $D\times\Omega$, where $D$ is a bounded open subset of $\mathbb{R}^{d}$ $\left(d>1\right)$, and $\left(\Omega,\tciFourier,P\right)$ is a complete probability space.

2. Weighted Stochastic Field Exponent Lebesgue and Sobolev Spaces

Definition 1**.**

We denote the family of all measurable functions $p\left(.,.\right):D\times\Omega\longrightarrow\left[1,\infty\right)$ (called a stochastic field exponent). In this paper, the function $p\left(.,.\right)$ always denotes a stochastic field exponent. Moreover, we put

[TABLE]

A positive, measurable and locally integrable function $\vartheta$ defined on $D\times\Omega$ is called a weight function. Now, we introduce the integrability conditions used on the framework of weighted variable Lebesgue and Sobolev spaces

[TABLE]

where $s\left(.,.\right)$ is a positive function. The weighted modular function $\rho_{p(.,.),\vartheta}$ on $D\times\Omega$ is defined by

[TABLE]

where $d\lambda=d\lambda\left(x,t\right)=dxdt$.

The spaces $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ consist of all measurable stochastic fields (functions) $u$ on $D\times\Omega$ such that $\mathop{\displaystyle\int}\limits_{D\times\Omega}\left|u(x,t)\right|^{p(x,t)}\vartheta(x,t)d\lambda<\infty$ and endowed with the Luxemburg norm

[TABLE]

It is well known that $u\in L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ if and only if $\left\|u\right\|_{p(.,.),\vartheta}=\left\|u\vartheta^{\frac{1}{p(.,.)}}\right\|_{p(.,.)}<\infty$. Moreover, it is clear that, if the inequality $0<C\leq\vartheta$ is satisfied, then $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)\hookrightarrow L^{p(.,.)}\left(D\times\Omega\right).$ In this study, we assume that $1<p^{-}\leq p(.,.)\leq p^{+}<\infty$ and $(H_{1}),(H_{2}).$ Moreover, we use the abbreviations and symbols; a.e., $\longrightarrow$ and $\rightharpoonup$ for almost everywhere, strong convergence and weak convergence, respectively.

It can be seen that the space $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ is uniformly convex, so it is reflexive, see [4]. Moreover, we denote by $L_{\vartheta^{\ast}}^{q(.,.)}\left(D\times\Omega\right)$ as the dual space of $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ where $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1$ and $\vartheta^{\ast}=\vartheta^{1-q(.,.)}.$

Now, we give the relationships between $\left\|.\right\|_{p(.,.),\vartheta}$ and $\rho_{p(.,.),\vartheta}$ as follows.

Proposition 1**.**

If $u\in L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$, then we have

1. (i)

$\left\|u\right\|_{p(.,.),\vartheta}^{p^{-}}\leq\rho_{p(.,.),\vartheta}\left(u\right)\leq\left\|u\right\|_{p(.,.),\vartheta}^{p^{+}}$* with $\left\|u\right\|_{p(.,.),\vartheta}\geq 1$.* 2. (ii)

$\left\|u\right\|_{p(.,.),\vartheta}^{p^{+}}\leq\rho_{p(.,.),\vartheta}\left(u\right)\leq\left\|u\right\|_{p(.,.),\vartheta}^{p^{-}}$* with $\left\|u\right\|_{p(.,.),\vartheta}\leq 1.$*

Theorem 1**.**

The inequality

[TABLE]

holds for every $f\in L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ and $g\in L_{\vartheta^{\ast}}^{q(.,.)}(D\times\Omega)$ with the constant $C$ depends on $p(.,.)$ where $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1$ and $\vartheta^{\ast}=\vartheta^{1-q(.)}$

Proof.

If we consider the Hölder inequality, then we get

[TABLE]

for some $C>0$. That is the desired result.

Theorem 2**.**

(see [12],[13])The space $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ is a reflexive Banach space with respect to norm $\left\|.\right\|_{p(.,.),\vartheta}$.

Proposition 2**.**

The space $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ is continuously embedded in $L_{loc}^{1}\left(D\times\Omega\right)$. This means that every function in $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ has distributional (weak) derivative.

Proof.

Suppose that $u\in L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ and let $K=K_{1}\times K_{2}\subset D\times\Omega$ be a compact set. By the Hölder inequality, there exists an $A_{K}>0$ such that

[TABLE]

where $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1.$ It is obvious that $\left\|\vartheta^{-\frac{1}{p(.,.)}}\right\|_{q(.,.),K}<\infty$ if and only if $\rho_{q(.,.),K}\left(\vartheta^{-\frac{1}{p(.,.)}}\right)<\infty.$ Since $\vartheta^{-\frac{1}{p(.,.)-1}}\in L_{loc}^{1}\left(D\times\Omega\right),$ we have

[TABLE]

If we use (2.1) and (2.2), then the proof is completed.

Remark 1**.**

If $\vartheta^{-\frac{1}{p(.,.)-1}}\notin L_{loc}^{1}\left(D\times\Omega\right)$, then the space $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ might not be continuously embedded in $L_{loc}^{1}\left(D\times\Omega\right)$.

Theorem 3**.**

Let $u\in L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ and $u_{n}\in L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ with $\left\|u_{n}\right\|_{p(.,.),\vartheta}\leq C$ for some $C>0$. If $u_{n}\longrightarrow u$ a.e. in $D\times\Omega$, then $u_{n}\rightharpoonup u$ in $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$.

Proof.

Since the space $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ is reflexive by Theorem 2, we only need to show that

[TABLE]

for each $g\in L_{\vartheta^{\ast}}^{q(.,.)}(D\times\Omega)$ where $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1$ and $\vartheta^{\ast}=\vartheta^{1-q(.,.)}.$ It is well known that $\left\|u_{n}\right\|_{p(.,.),\vartheta}\leq C$ if and only if $\rho_{p(.,.),\vartheta}\left(\frac{u_{n}}{C}\right)\leq 1$ for each $n\in\mathbb{N}$. This follows by the Fatou’s Lemma and the definition of the norm that

[TABLE]

Thus, we get $\left\|u\right\|_{p(.,.),\vartheta}\leq C$. By the absolute continuity of the Lebesgue integral, we have

[TABLE]

where $g\in L_{\vartheta^{\ast}}^{q(.,.)}(D\times\Omega)$ and $K\subset D\times\Omega$. This yields that $\underset{meas(K)\longrightarrow 0}{\lim}\left\|g\chi_{K}\right\|_{q(.,.),\vartheta^{\ast}}=0$, and there exists a $\delta>0$ such that

[TABLE]

for $meas(K)<\delta.$ By the Egorov theorem, there exists a set $L\subset D\times\Omega$ such that $u_{n}\longrightarrow u$ uniformly on $L$ with $meas\left(\left(D\times\Omega\right)-L\right)<\delta$. If we choose $n_{0}$ such that $n\geq n_{0}$, then we have

[TABLE]

Let us denote $K=\left(D\times\Omega\right)-L.$ By (2.3) and (2.4), we have

[TABLE]

That is the desired result.

Definition 2**.**

We set the weighted stochastic field variable Sobolev spaces $W_{\vartheta}^{k,p(.,.)}\left(D\times\Omega\right)$ by

[TABLE]

equipped with the norm

[TABLE]

where $\alpha\in\mathbb{N}_{0}^{d}$ is a multi-index, $\left|\alpha\right|=\alpha_{1}+\alpha_{2}+...+\alpha_{d}$ and $D^{\alpha}=\frac{\partial^{\left|\alpha\right|}}{\partial_{x_{1}}^{\alpha_{1}}...\partial_{x_{d}}^{\alpha_{d}}}$. The space $\left(W_{\vartheta}^{k,p(.,.)}\left(D\times\Omega\right),\left\|.\right\|_{W_{\vartheta}^{k,p(.,.)}\left(D\times\Omega\right)}\right)$ is a reflexive Banach space by [13, Theorem 2.5].

Moreover, the space $W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ is defined by

[TABLE]

with the norm $\left\|u\right\|_{W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)}=\left\|u\right\|_{p(.,.),\vartheta}+\left\|\nabla u\right\|_{p(.,.),\vartheta}$.

The space $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ is the closure of

[TABLE]

in $W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$. Also, it is obvious that $C\left(D\times\Omega\right)$ is a subspace of $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$, and the dual space of $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ is $W_{0,\vartheta^{\ast}}^{-1,q(.,.)}\left(D\times\Omega\right)$, where $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1$ and $\vartheta^{\ast}=\vartheta^{1-q(.,.)}$.

The following theorem present the Poincaré inequality for the weighted Sobolev spaces $W_{0,\vartheta}^{k,p(.,.)}\left(D\times\Omega\right)$. For the proof, we refer [14].

Remark 2**.**

If we use the similar method in [14, Theorem 7], then we have the Poincaré inequality for $W_{0,\vartheta}^{k,p(.,.)}\left(D\times\Omega\right),$ that is, there exists a $C>0$ such that the inequality

[TABLE]

holds for every $u\in W_{0,\vartheta}^{k,p(.,.)}\left(D\times\Omega\right)$ (or $u\in C\left(D\times\Omega\right)$).

Therefore, the space $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ equipped with the norm

[TABLE]

for $u\in W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right).$ It is note that the norms $\left\|.\right\|_{W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)}$ and $\left\|\left|.\right|\right\|_{W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)}$ are equivalent on $W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right).$ Then, $W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ is continuously embedded in $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$ if and only if the inequality (2.5) is satisfied for every $u\in W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right).$

3. Compact Embedding Theorems

In this section, we present several compact embeddings between the weighted stochastic field variable Lebesgue and Sobolev spaces. Because, we need these embeddings to investigate weak solutions of stochastic partial differential equation (1.1). Now, let us introduce the function $p^{\ast}\left(.,.\right)$ and $p_{s}\left(.,.\right)$ defined by

[TABLE]

[TABLE]

and we have

[TABLE]

for almost all $(x,t)\in D\times\Omega$.

Proposition 3**.**

(see [9])Assume that the boundary of $D\times\Omega$ possesses the cone property and $p\left(.,.\right)\in C\left(\overline{D\times\Omega}\right)$. If $q(.,.)\in C\left(\overline{D\times\Omega}\right)$ and $1\leq q(x,t)\leq p^{\ast}\left(x,t\right)$ for $(x,t)\in\overline{D\times\Omega}$, then $W^{1,p(.,.)}\left(D\times\Omega\right)$ is compactly embedded in $L^{q(.,.)}\left(D\times\Omega\right)$.

Theorem 4**.**

Assume that the boundary of $D\times\Omega$ possesses the cone property, $p(.,.)\in C\left(\overline{D\times\Omega}\right)$ and $1<p(x,t)$ for all $(x,t)\in\overline{D\times\Omega}$. Suppose that

1. (i)

$0<\vartheta(x,t)\in L^{\alpha(.,.)}\left(D\times\Omega\right)$* with $(x,t)\in D\times\Omega$, $\alpha(.,.)\in C\left(\overline{D\times\Omega}\right)$ and $1<\alpha^{-}$.* 2. (ii)

$1<q(x,t)<\frac{\alpha(x,t)-1}{\alpha(x,t)}p^{\ast}\left(x,t\right)$* for all $(x,t)\in\overline{D\times\Omega}$.*

Then, there is a compact embedding from $W^{1,p(.,.)}\left(D\times\Omega\right)$ to $L_{\vartheta}^{q(.,.)}(D\times\Omega).$

Proof.

For the proof, we use similar method in [5, Theorem 2.1]. Let $u\in W^{1,p(.,.)}\left(D\times\Omega\right)$ and set $r(x,t)=\frac{\alpha(x,t)}{\alpha(x,t)-1}q(x,t)=\alpha_{0}(x,t)q(x,t)$. Then (ii) implies $r(x,t)<p^{\ast}\left(x,t\right).$ This follows that $W^{1,p(.,.)}\left(D\times\Omega\right)\hookrightarrow\hookrightarrow L^{r(.,.)}\left(D\times\Omega\right)$ by Proposition 3. Moreover, for $u\in W^{1,p(.,.)}\left(D\times\Omega\right),$ we get $\left|u\right|^{q(x,t)}\in L^{\alpha_{0}(x,t)}\left(D\times\Omega\right)$. By the Hölder inequality

[TABLE]

Therefore, we have $W^{1,p(.,.)}\left(D\times\Omega\right)\subset L_{\vartheta}^{q(.,.)}(D\times\Omega).$ Now, let $\left(u_{n}\right)_{n\in\mathbb{N}}\subset W^{1,p(.,.)}\left(D\times\Omega\right)$ and $u_{n}\rightharpoonup 0$ in $W^{1,p(.,.)}\left(D\times\Omega\right)$. Since $W^{1,p(.,.)}\left(D\times\Omega\right)$ is compactly embedded in $L^{r(.,.)}\left(D\times\Omega\right)$ by the Proposition 3, we get $u_{n}\longrightarrow 0$ in $L^{r(.,.)}\left(D\times\Omega\right)$, that is, $\left\|u_{n}\right\|_{r(.,.)}\longrightarrow 0.$ This yields $\rho_{r(.,.)}(u_{n})\longrightarrow 0$ and

[TABLE]

or equivalently

[TABLE]

Thus, we have

[TABLE]

which implies $\left\|u_{n}\right\|_{q(.,.),\vartheta}\longrightarrow 0.$ That is the desired result.

Theorem 5**.**

Let all conditions in Proposition 3 be hold. Moreover, assume that the assumptions in Theorem 4 replacing $p\left(.,.\right)$ by $p_{s}\left(.,.\right)$ are also satisfied. Thus we obtain

[TABLE]

where $r(x,t)\leq p_{s}^{\ast}\left(x,t\right)$ for all $\left(x,t\right)\in D\times\Omega$ and $0<C\leq\vartheta$.

Proof.

First of all, we will show that $W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ is continuously embedded in $W^{1,p_{s}(.,.)}\left(D\times\Omega\right)$. Let $u\in W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$. Then it is clear that $u,\left|\nabla u\right|\in L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)$. If we consider the Hölder inequality, Proposition 2 and $\vartheta^{-s(.,.)}\in L^{1}\left(D\times\Omega\right)$, then we have

[TABLE]

where

[TABLE]

and

[TABLE]

Hence, we get

[TABLE]

where

[TABLE]

and

[TABLE]

Therefore, we obtain

[TABLE]

Since $p_{s}\left(.,.\right)<p(.,.)$, we have $L_{\vartheta}^{p(.,.)}\left(D\times\Omega\right)\hookrightarrow L^{p(.,.)}\left(D\times\Omega\right)\hookrightarrow L^{p_{s}(.,.)}\left(D\times\Omega\right)$, see [7, Theorem 2.8]. Then there exists $C^{\ast\ast}>0$ such that

[TABLE]

for almost everywhere in $D\times\Omega$. By (3.3) and (3.4), we conclude that $W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)\subset W^{1,p_{s}(.,.)}\left(D\times\Omega\right)$. If we consider the Banach theorem in [3], we get $W_{\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)\hookrightarrow W^{1,p_{s}(.,.)}\left(D\times\Omega\right)$. This follows from Theorem 4 that

[TABLE]

This completes the proof.

Now, we reveal some required conditions for the equation (1.1). Assume that $A:\mathbb{R}^{d}\times\Omega\times\mathbb{R}\times\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$ and $f:\mathbb{R}^{d}\times\Omega\longrightarrow\mathbb{R}$ satisfy the following growth conditions:

1. $\left(H_{3}\right)$

$\left|A\left(x,t,s,\xi\right)\right|\leq\beta\vartheta^{\frac{1}{p(x,t)}}\left[k(x,t)+\vartheta^{\frac{1}{q(x,t)}}\left|\xi\right|^{p(x,t)-1}\right]$ 2. $\left(H_{4}\right)$

$\left(A\left(x,t,s,\xi\right)-A\left(x,t,s,\mu\right)\right)\left(\xi-\mu\right)>0$, $\xi\neq\mu$ 3. $\left(H_{5}\right)$

$A\left(x,t,s,\xi\right)\left|\xi\right|\geq\alpha\vartheta(x,t)\left|\xi\right|^{p(x,t)}$

where $k(x,t)$ is a positive function in $L^{q(.,.)}(D\times\Omega)$ where $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1$ and $\alpha$, $\beta$ are positive constants.

Let $A_{0}\left(x,t,s,\xi\right):\mathbb{R}^{d}\times\Omega\times\mathbb{R}\times\mathbb{R}^{d}\longrightarrow\mathbb{R}$ be a Carathéodory function such that for a.e. $\left(x,t\right)\in\mathbb{R}^{d}\times\Omega$ and for all $s\in\mathbb{R}$, $\xi\in\mathbb{R}^{d}$, the growth condition

[TABLE]

is satisfied where $g:\mathbb{R}\longrightarrow\mathbb{R}^{+}$ is a continuous function that belongs to $L^{1}\left(\mathbb{R}\right)$ and $\gamma\left(x,t\right)$ belongs to $L_{\vartheta^{\ast}}^{q\left(.,.\right)}\left(D\times\Omega\right)$. Finally, we assume that $f\in W_{\vartheta^{\ast}}^{-1,q(.,.)}\left(D\times\Omega\right).$

Lemma 1**.**

(see [8])Assume that $\left(H_{3}\right)-\left(H_{5}\right)$ hold and let $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ be a sequence in $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ such that $u_{n}\rightharpoonup u$ in $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ and

[TABLE]

Then $u_{n}\longrightarrow u$ in $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$.

4. Existence of Weak Solution of Stochastic Partial Differential

Equations With Stochastic Field Growth

Definition 3**.**

A function $u\in W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ is said to be a weak solution (1.1), if

[TABLE]

for all $\varphi\in W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$.

Definition 4**.**

A bounded operator $T$ from $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ to its dual $W_{\vartheta^{\ast}}^{-1,q(.,.)}\left(D\times\Omega\right)$ is called pseudo-monotone if and only if for any sequences $\left(u_{k}\right)_{k\in\mathbb{N}}$ in $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ satisfying

1. (i)

$u_{k}\rightharpoonup u$* in $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ as $k\longrightarrow\infty,$* 2. (ii)

$\limsup\limits_{k\longrightarrow\infty}\langle T\left(u_{k}\right),u_{k}-u\rangle\leq 0$**

imply $T\left(u_{k}\right)\rightharpoonup T\left(u\right)$ and $\langle T\left(u_{k}\right),u_{k}\rangle\longrightarrow\langle T\left(u\right),u\rangle.$

Definition 5**.**

Assume that $X$ is a reflexive Banach space and $X^{\ast}$ denotes dual of $X.$ Also, let $\langle.,.\rangle$ be a pair between $X$ and $X^{\ast}$. Then a mapping $\Gamma:X\longrightarrow X^{\ast}$ is called coercive if there exists a $u\in X$ such that

[TABLE]

Let us define the operator $\Gamma:W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)\longrightarrow W_{\vartheta^{\ast}}^{-1,q(.,.)}\left(D\times\Omega\right)$ by

[TABLE]

where $\varphi\in W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ and $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1$. Hence, we can write the equation (1.1) as $\langle\Gamma(u),\varphi\rangle=\langle f,\varphi\rangle.$

Proposition 4**.**

(Weak compactness of bounded set)Let $X$ be a reflexive Banach space. Moreover, assume that $\left(u_{k}\right)_{k\in\mathbb{N}}$ is a sequence such that

(i) $u_{k}\in X$

(ii) $\left\|u_{k}\right\|_{X}\leq C$ for all $k\in\mathbb{N},$

that is, $\left(u_{k}\right)_{k\in\mathbb{N}}$ is a bounded sequence in $X,$ then there exists a subsequence $\left(u_{k_{l}}\right)_{l\in\mathbb{N}}$ and an element $u_{0}\in X$ such that $u_{k_{l}}\rightharpoonup u_{0}$ in $X.$

Theorem 6**.**

(see [10])Let X be a reflexive Banach space and assume $\Gamma:X\longrightarrow X^{\ast}$ is continuous (bounded), coercive and pseudo-monotone. Then for every $g\in X^{\ast}$ there exists a solution $u\in X$ of the equation $\Gamma\left(u\right)=g.$

Now, we are ready to give our main motivation of the paper.

Theorem 7**.**

If the conditions $\left(H_{3}\right),$ $\left(H_{4}\right)$ and $\left(H_{5}\right)$ hold, then there exists at least a weak solution of (1.1).

Proof.

The proof base on three parts.

**Step 1. **First of all, we will show that the operator $\Gamma$ is bounded. The operator $\Gamma$ is equal to the sum of two operators such that $\Gamma=\Gamma_{1}+\Gamma_{2}$ where

[TABLE]

and

[TABLE]

If we consider $\left(H_{3}\right)$, Proposition 1 and Hölder inequality, then we have

[TABLE]

where

[TABLE]

and $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1$. This yields that $\Gamma_{1}$ is bounded. In similar way, since $\gamma\in L_{\vartheta^{\ast}}^{q\left(.,.\right)}\left(D\times\Omega\right),$ we get

[TABLE]

[TABLE]

where

[TABLE]

and $\frac{1}{p(.,.)}+\frac{1}{q(.,.)}=1$. Thus, we obtain that $\Gamma_{2}$ is bounded. Therefore, we get that $\Gamma$ is bounded.

Step 2. Now, we will show that the operator $\Gamma$ is coercive. By $\left(H_{5}\right)$, we have

[TABLE]

for some $r>1.$ On the other hand, since the norm $\left\|A_{0}\left(x,t,u,\nabla u\right)\right\|_{q(.,.),\vartheta^{\ast}}$ is bounded, then we have

[TABLE]

This follows that

[TABLE]

**Step 3. **Now, we will obtain that the operator $\Gamma$ is pseudo-monotone from $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ to $W_{\vartheta^{\ast}}^{-1,q(.,.)}\left(D\times\Omega\right)$. Let $u_{k}\rightharpoonup u$ in $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ and $\limsup\limits_{k\longrightarrow\infty}\langle\Gamma\left(u_{k}\right),u_{k}-u\rangle\leq 0$. Since $\Gamma$ is bounded and $u_{k}\rightharpoonup u,$ then we have

[TABLE]

By (4.2), we can write that

[TABLE]

By the growth condition $\left(H_{3}\right)$ and Proposition 4, the sequence $\left(A\left(x,t,u_{k},\nabla u_{k}\right)\right)_{k\in\mathbb{N}}$ is bounded in $\left(L_{\vartheta^{\ast}}^{q(.,.)}(D\times\Omega)\right)^{d}$ such that

[TABLE]

as $k\longrightarrow\infty.$ Similarly, since $\left(A_{0}\left(x,t,u_{k},\nabla u_{k}\right)\right)_{k}$ is bounded in $L_{\vartheta^{\ast}}^{q(.,.)}(D\times\Omega)$, then there exists a function $\psi\in L_{\vartheta^{\ast}}^{q(.,.)}(D\times\Omega)$ such that

[TABLE]

as $k\longrightarrow\infty.$ For all $v\in W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right),$ we have

[TABLE]

Since $W_{0,\vartheta}^{1,p(.,.)}\left(D\times\Omega\right)$ is compactly embedded in $L_{\vartheta}^{p(.,.)}(D\times\Omega)$ by Theorem 5, we get

[TABLE]

By (4.5) and (4.7), we have

[TABLE]

as $k\longrightarrow\infty.$ On the other hand, if we consider (4.3) and (4.6), then we have

[TABLE]

Hence we obtain

[TABLE]

Due to $\left(H_{4}\right)$, we have

[TABLE]

and

[TABLE]

Moreover, by (4.4), we get

[TABLE]

It is obtained that

[TABLE]

by (4.9). If we consider (4.6), (4.8) and (4.10), then we have

[TABLE]

This follows from (4.7) and $\left(H_{3}\right)$ that

[TABLE]

in $\left(L_{\vartheta^{\ast}}^{q(.,.)}(D\times\Omega)\right)^{d}$. Hence, we get

[TABLE]

By Lemma 1, we obtain

[TABLE]

and then $\nabla u_{k}\longrightarrow\nabla u$ a.e. in $D\times\Omega$ for a subsequence denoted by $\left(u_{k}\right)_{k\in\mathbb{N}}$. Since $A$ and $A_{0}$ are Carathéodory functions, we have

[TABLE]

This yields that $h=\Gamma\left(u\right)$ and the operator $\Gamma$ is pseudo-monotone. Finally, if we consider the Theorem 6, then there exists at least a weak solution of (1.1).