Study on the generalized $(p,q)$-Laplacian elliptic systems, parabolic systems and integro-differential systems

Abstract

In this paper, we present the abstract results for the existence and uniqueness of the solution of nonlinear elliptic systems, parabolic systems and integro-differential systems involving the generalized $(p,q)$-Laplacian operator. Our method makes use of the characteristics of the ranges of linear and nonlinear maximal monotone operators and the subdifferential of a proper, convex, and lower-semi-continuous functional, and we employ some new techniques in the construction of the operators and in proving the properties of the newly defined operators. The systems discussed in this paper and the method used extend and complement some of the previous work.

Introduction and preliminaries

Introduction

Nonlinear boundary value problems involving the generalized p-Laplacian operator arise from many physical phenomena, such as reaction-diffusion problems, petroleum extraction, flow through porous media and non-Newtonian fluids, just to name a few. Hence, the study of such problems and their generalizations have attracted numerous attention in recent years. For example, based on Calvert and Gupta’s [1] result on perturbations of the ranges of m-accretive mappings (stated as Theorem 1.1 in Section 1.2), Wei and Agarwal [2] have studied the following nonlinear elliptic boundary value problem involving the generalized p-Laplacian:

$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div} [(C(x)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ]+\varepsilon|u|^{q-2}u+g(x,u(x))= f(x),\quad \mbox{a.e. in } \Omega, \\ - \langle\vartheta,(C(x)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u \rangle\in\beta_{x}(u(x)),\quad \mbox{a.e. on } \Gamma, \end{array}\displaystyle \right . $$

(1.1)

where $0\leq C(x)\in L^{p}(\Omega)$, $\beta_{x}$ is the subdifferential of a proper, convex, and lower-semi-continuous function, ε is a non-negative constant and ϑ denotes the exterior normal derivative of Γ. It is shown that (1.1) has solutions in $L^{s}(\Omega)$ under some conditions, where $\frac{2N}{N+1} < p\leq s <+\infty$, $1\leq q<+\infty$ if $p\geq N$, and $1\leq q \leq \frac{Np}{N-p}$ if $p< N$, for $N \geq1$.

Recently, the work on the generalized p-Laplacian operator problem (1.1) is extended to the so-called p-Laplacian-like problem

$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div} [(C(x)+|\nabla u|^{2})^{\frac{s}{2}}|\nabla u|^{m-1}\nabla u ] + \varepsilon|u|^{q-2}u + g(x,u(x)) = f(x),\quad \mbox{in } \Omega, \\ - \langle\vartheta,(C(x)+|\nabla u|^{2})^{\frac {s}{2}}|\nabla u|^{m-1}\nabla u \rangle \in\beta_{x}(u), \quad \mbox{on } \Gamma. \end{array}\displaystyle \right . $$

(1.2)

Using Theorem 1.1 again, it is shown in [3] that (1.2) has solutions in $L^{p}(\Omega)$ under some conditions, where $\frac{2N}{N+1} < p <+\infty$, $1\leq q<+\infty$ if $p\geq N$, and $1\leq q \leq\frac{Np}{N-p}$ if $p< N$, for $N \geq1$.

Since one system, expressed by one equation, interacts with another system in reality, the study of nonlinear systems with $(p,q)$-Laplacian is also an important topic. In the non-Newtonian theory, the quantity $(p,q)$ is a characteristic of the medium. Media with $(p,q)> (2,2)$ are called dilatant fluids, those with $(p,q) < (2,2)$ are called pseudodoplastics, and if $(p,q) = (2,2)$, they are called Newtonian fluids. The studies on the p-Laplacian boundary value problems have been extended to cases of nonlinear Neumann elliptic systems with $(p,q)$-Laplacian. For example, in [4] the following system with Neumann boundaries has been discussed:

$$ \left \{ \textstyle\begin{array}{l} -\Delta_{p}u + \varepsilon_{1} |u|^{p-2}u+g(x,u(x),v(x)) = f_{1}(x),\quad \mbox{a.e. in } \Omega, \\ -\Delta_{q}v+ \varepsilon_{2} |v|^{q-2}v + g(x,v(x),u(x)) = f_{2}(x),\quad \mbox{a.e. in } \Omega, \\ - \langle\vartheta, |\nabla u|^{p-2}\nabla u \rangle \in \beta_{x}(u(x)) ,\quad \mbox{a.e. on } \Gamma, \\ - \langle\vartheta, |\nabla v|^{q-2}\nabla v \rangle\in \beta_{x}(v(x)),\quad \mbox{a.e. on } \Gamma. \end{array}\displaystyle \right . $$

(1.3)

Inspired by Theorem 1.1 again, a sufficient condition on the existence of a solution in $L^{p}(\Omega)\times L^{q}(\Omega)$ is presented in [4].

On the other hand, based on Brezis’ result [5] (stated as Theorem 1.2 in Section 1.2), Wei et al. [6] have studied the following nonlinear Dirichlet elliptic system in $W^{1,p}(\Omega)\times W^{1,q}(\Omega)$:

$$ \left \{ \textstyle\begin{array}{l} -\Delta_{p}u+\varepsilon_{1}|u|^{p-2}u-\Delta_{q} v+\varepsilon_{2}|v|^{q-2}v =f_{1}(x)+f_{2}(x),\quad \mbox{a.e. in } \Omega, \\ \gamma_{1} u = g_{1}(x) , \qquad \gamma_{2} v = g_{2}(x),\quad \mbox{a.e. on } \Gamma, \end{array}\displaystyle \right . $$

(1.4)

and then extend (1.4) to the following two cases with generalized $(p,q)$-Laplacian:

$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div}(\alpha_{1}(\operatorname{grad} u))+\varepsilon_{1}|u|^{p-2}u-\operatorname{div}(\alpha_{2}(\operatorname{grad} v))+\varepsilon_{2}|v|^{q-2}v \\ \quad =f_{1}(x)+f_{2}(x), \quad \mbox{a.e. in } \Omega, \\ \gamma_{1} u = g_{1}(x) ,\qquad \gamma_{2} v = g_{2}(x), \quad \mbox{a.e. on } \Gamma \end{array}\displaystyle \right . $$

(1.5)

and

$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div} [(C_{1}(x)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ]+\varepsilon_{1}|u|^{p-2}u -\operatorname{div} [(C_{2}(x)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ]+ \varepsilon_{2}|v|^{q-2}v \\ \quad =f_{1}(x)+f_{2}(x),\quad \mbox{a.e. in } \Omega, \\ \gamma_{1} u = g_{1}(x) ,\qquad \gamma_{2} v = g_{2}(x),\quad \mbox{a.e. on } \Gamma. \end{array}\displaystyle \right . $$

(1.6)

Integro-differential equation is also a much-studied topic in applied mathematics. Most of the existing techniques used to discuss the existence and uniqueness of the solution to integro-differential equation involves the finite element method. In [7], a new method based on a result of Zeidler [8] (stated as Theorem 1.3 in Section 1.2) is employed to tackle the following nonlinear integro-differential equation involving the generalized p-Laplacian operator with mixed boundary conditions:

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u}{\partial t}-\operatorname{div} [(C(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] +\varepsilon|u|^{q-2}u + a \frac{\partial}{\partial t}\int_{\Omega}u \,dx \\ \quad = f(x,t),\quad (x,t)\in\Omega\times(0,T), \\ - \langle\vartheta, (C(x,t)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u \rangle\in\beta_{x}(u) ,\quad (x,t)\in\Gamma\times (0,T), \\ u(x,0) = u(x,T), \quad x \in \Omega. \end{array}\displaystyle \right . $$

(1.7)

It is proved that (1.7) has a unique solution in $L^{p}(0 , T; W^{1,p}(\Omega))$, where $1 < q \leq p < +\infty$.

Inspired by the work on (1.7), the following nonlinear integro-differential system involving the generalized $(p,q)$-Laplacian is investigated in [9]:

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u(x,t)}{\partial t} -\operatorname{div} [(C_{1}(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] + \varepsilon_{1} |u|^{r-2}u + g_{1}(x,u, \nabla u)+ a_{1} \frac{\partial}{\partial t} \int_{\Omega}u\,dx \\ \quad = f_{1}(x,t),\quad (x,t) \in\Omega\times(0,T), \\ \frac{\partial v(x,t)}{\partial t} - \operatorname{div} [(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ] + \varepsilon_{2} |v|^{s-2}v + g_{2}(x,v,\nabla v)+ a_{2} \frac{\partial}{\partial t} \int_{\Omega}v\,dx \\ \quad =f_{2}(x,t),\quad (x,t) \in\Omega\times(0,T), \\ - \langle\vartheta,(C_{1}(x,t)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u \rangle\in\beta_{x}(u),\quad (x,t) \in\Gamma\times(0,T), \\ - \langle\vartheta,(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v \rangle\in \beta_{x}(v),\quad (x,t) \in\Gamma\times(0,T), \\ u(x,0) = u(x, T), \qquad v(x,0) = v(x,T),\quad x \in\Omega, \end{array}\displaystyle \right . $$

(1.8)

where $\nabla u = (\frac{\partial u}{\partial x_{1}}, \frac {\partial u}{\partial x_{2}}, \ldots, \frac{\partial u}{\partial x_{N}} )$ and $x = (x_{1}, x_{2}, \ldots, x_{N}) \in\Omega$. Based on a result of [10] (stated as Theorem 1.4 in Section 1.2), the existence of the unique non-trivial solution of (1.8) in $L^{p}(0 , T;W^{1,p}(\Omega))\times L^{q}(0 , T; W^{1,q}(\Omega))$ is presented, where $N\geq1$, $\frac{2N}{N+1} < r \leq\min\{p,p'\} < +\infty$, and $\frac{2N}{N+1} < s \leq\min\{q,q'\} < +\infty$. (Here, $\frac{1}{p}+\frac{1}{p'}=1$, $\frac{1}{q}+\frac{1}{q'}=1$.)

Parabolic equations are equally important as elliptic equations and integro-differential equations. The generalized $(p,q)$-Laplacian parabolic equation with mixed boundaries has been extensively studied in [11],

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u}{\partial t}-\operatorname{div} [(C(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] + \varepsilon|u|^{p-2}u = f(x,t),\quad (x,t) \in\Omega\times(0,T), \\ - \langle\vartheta,(C(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u \rangle\in\beta(u) - h(x,t),\quad (x,t) \in\Gamma\times (0,T), \\ u(x,0) = u(x,T),\quad x \in\Omega. \end{array}\displaystyle \right . $$

(1.9)

It is shown that (1.9) has a unique solution in $L^{p}(0,T; W^{1,p}(\Omega))$ where $p \geq2$. The discussion of (1.9) in [11] is mainly based on Theorem 1.2 and a result of Reich [12] (stated as Theorem 1.5 in Section 1.2).

From the above research, we notice that it is not easy to check the assumptions presented in Theorems 1.1-1.5. As such we are motivated to extend the previous work to new problems and also to simplify the proof of the result. Indeed, motivated by the systems (1.4)-(1.6), (1.8), and (1.9), in this paper we shall employ a result of Zeidler [8] (stated as Theorem 1.6 in Section 1.2) as the main tool to obtain sufficient conditions for the existence and uniqueness of solutions for three nonlinear systems - the first is a nonlinear elliptic system involving the generalized $(p,q)$-Laplacian with Neumann boundaries, the second is a nonlinear parabolic system involving the generalized $(p,q)$-Laplacian with mixed boundaries, and the third is a nonlinear integro-differential system involving the generalized $(p,q)$-Laplacian with mixed boundaries. The three systems considered are as follows:

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} -\operatorname{div} [(C_{1}(x)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u ] + \varepsilon_{1} |u|^{r-2}u - \operatorname{div} [(C_{2}(x)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ] \\ \qquad {}+ \varepsilon_{2} |v|^{s-2}v + g_{1}(x, u , \nabla u)+ g_{2}(x,v,\nabla v) \\ \quad = f_{1}(x)+ f_{2}(x),\quad x \in\Omega, \\ - \langle\vartheta,(C_{1}(x)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u \rangle- \langle\vartheta,(C_{2}(x)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v \rangle \\ \quad \in \beta_{x}(u)+\beta_{x}(v),\quad x \in\Gamma; \end{array}\displaystyle \right . \end{aligned}$$

(1.10)

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u(x,t)}{\partial t} -\operatorname{div} [(C_{1}(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] + \varepsilon_{1} |u|^{r-2}u + g_{1}(x,u, \nabla u)+\frac{\partial v(x,t)}{\partial t} \\ \qquad {} - \operatorname{div} [(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ] + \varepsilon_{2} |v|^{s-2}v + g_{2}(x,v,\nabla v) \\ \quad = f_{1}(x,t) +f_{2}(x,t),\quad (x,t) \in\Omega\times(0,T), \\ - \langle\vartheta,(C_{1}(x,t)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u \rangle- \langle\vartheta,(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v \rangle \\ \quad \in\beta_{x}(u)+ \beta_{x}(v),\quad (x,t) \in\Gamma\times (0,T), \\ u(x,0) = u(x, T), \qquad v(x,0) = v(x,T), \quad x \in\Omega; \end{array}\displaystyle \right . \end{aligned}$$

(1.11)

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u(x,t)}{\partial t} -\operatorname{div} [(C_{1}(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] + \varepsilon_{1} |u|^{r-2}u+ g_{1}(x,u, \nabla u) \\ \qquad {}+a_{1}\frac{\partial}{\partial t }\int_{\Omega}u \,dx+ \frac{\partial v(x,t)}{\partial t} - \operatorname{div} [(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ] \\ \qquad {} + \varepsilon_{2} |v|^{s-2}v + g_{2}(x,v,\nabla v)+a_{2}\frac{\partial}{\partial t }\int_{\Omega}v \,dx \\ \quad = f_{1}(x,t) +f_{2}(x,t), \quad (x,t) \in\Omega\times(0,T), \\ - \langle\vartheta,(C_{1}(x,t)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u \rangle- \langle\vartheta,(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v \rangle \\ \quad \in \beta_{x}(u)+\beta_{x}(v),\quad (x,t) \in \Gamma\times(0,T), \\ u(x,0) = u(x, T), \qquad v(x,0) = v(x,T), \quad x \in\Omega. \end{array}\displaystyle \right . \end{aligned}$$

(1.12)

The investigation of systems (1.10)-(1.12) will be presented in Sections 2-4, respectively, and more details of these systems will be introduced in these sections. Finally, in Section 5 we shall present some examples of (1.10)-(1.12).

Preliminaries

Let X be a real Banach space with its dual $X^{*}$ being strictly convex. We shall use $(\cdot,\cdot)$ to denote the generalized duality pairing between X and $X^{*}$. For any subset G of X, we denote by intG its interior and G̅ its closure, respectively. For two subsets $G_{1}$ and $G_{2}$ in X, if $\overline{G}_{1}= \overline{G}_{2}$ and $\operatorname{int} G_{1} = \operatorname{int} G_{2}$, then we say that $G_{1}$ is almost equal to $G_{2}$, denoted by $G_{1} \simeq G_{2}$. We use ‘$w\mbox{-}\!\lim$’ to denote the weak convergence. A mapping $T:D(T)= X\rightarrow X^{*}$ is said to be hemi-continuous on X [13] if $w\mbox{-}\!\lim_{t \rightarrow 0}T(x+ty) = Tx$, for any $x,y \in X$. A mapping $T:D(T)= X\rightarrow X^{*}$ is said to be demi-continuous on X [13] if $w\mbox{-}\!\lim_{n \rightarrow\infty}Tx_{n} = Tx$, for any sequence $\{x_{n}\}$ strongly converges to x in X.

Let $J_{r}$ denote the duality mapping from X into $2^{X^{*}}$, which is defined by

$$J_{r}(x)=\bigl\{ f\in X^{*} : (x, f) = \|x\|^{r}, \|f\|=\|x\|^{r-1}\bigr\} ,\quad x \in X, $$

where $r > 1$ is a constant. If $r \equiv2$, then we use J to denote $J_{2}$, which is called the normalized duality mapping. It is well known that, in general, $J_{r}(x) = \|x\|^{r-2}J(x)$, for all $x \neq0$. Since $X^{*}$ is strictly convex, J is a single-valued mapping [1, 14].

A multi-valued mapping $B:X\rightarrow2^{X^{*}}$ is said to be monotone [14] if $(u_{1}-u_{2},w_{1}-w_{2})\ge0$, for any $u_{i}\in D(B)$ and $w_{i}\in Bu_{i}$, $i=1,2$. The monotone operator B is said to be maximal monotone if $R(J+rB) = X^{*}$, for any $r> 0$. The mapping $B:X\rightarrow2^{X^{*}}$ is said to be strictly monotone [14] if $(u_{1}-u_{2},w_{1}-w_{2})= 0$, for $w_{i}\in Bu_{i}$, $i=1,2$, implies $u_{1} = u_{2}$. The mapping B is said to be coercive [13, 14] if $\lim_{n\rightarrow +\infty}{(x_{n},x^{*}_{n})}/{\|x_{n}\|}=+\infty$ for all $x_{n}\in D(B)$, $x^{*}_{n}\in Bx_{n}$ such that $\lim_{n\rightarrow +\infty}\|x_{n}\|= +\infty$.

Let $B : X \rightarrow2^{X^{*}}$ be a maximal monotone operator such that $0 \in B0$, then the equation $J(u_{t} - u) + t Bu_{t} \ni 0$ has a unique solution $u_{t} \in D(B)$ for every $u \in X$ and $t > 0$. The resolvent $J_{t}^{B}$ and the Yosida approximation $B_{t}$ of B are defined by $J_{t}^{B} u = u_{t}$ and $B_{t} u = -\frac{1}{t}J(u_{t}-u)$ for all $u \in X$ and $t > 0$ [14].

For $k \in(-\infty, +\infty)$, a multi-valued mapping $\widetilde{A}: D(\widetilde{A})\subset X \rightarrow2^{X}$ is said to be k-accretive [10] if

$$ \bigl(v_{1}-v_{2},J(u_{1}-u_{2}) \bigr)\geq k \|u_{1}-u_{2}\|^{2}, $$

(1.13)

for any $u_{i} \in D(\widetilde{A})$ and $v_{i}\in\widetilde{A}u_{i}$, $i=1,2$. For $k > 0$ in inequality (1.13), we say that à is strongly accretive while for $k = 0$, à is simply called accretive. An accretive mapping à is said to be m-accretive if $R(I+\lambda\widetilde{A})=X$ for some $\lambda>0$. We say that a mapping $\widetilde{A}:X\rightarrow2^{X}$ is boundedly-inversely-compact [1] if, for any pair of bounded subsets G and $G'$ of X, the subset $G\cap \widetilde{A}^{-1}(G')$ is relatively compact in X.

Let C be a closed convex subset of X and let $A: C \rightarrow 2^{X^{*}}$be a multi-valued mapping. Then A is said to be a pseudo-monotone operator [14] provided that

• (i) for each $x \in C$, the image Ax is a non-empty closed and convex subset of $X^{*}$;

• (ii) if $\{x_{n}\}$ is a sequence in C converging weakly to $x \in C$ and if $f_{n} \in Ax_{n}$ is such that $\limsup_{n\rightarrow \infty}(x_{n} - x,f_{n}) \leq0$, then to each element $y \in C$, there corresponds an $f(y) \in Ax$ with the property that $(x - y, f(y))\leq\liminf_{n \rightarrow\infty} (x_{n} - x, f_{n})$;

• (iii) for each finite-dimensional subspace F of X, the operator A is continuous from $C \cap F$ to $X^{*}$ in the weak topology.

A function Φ is called a proper convex function on X [14] if Φ is defined from X to $(-\infty, +\infty]$, not identically +∞, such that $\Phi((1-\lambda)x+\lambda y)\leq(1-\lambda)\Phi(x)+\lambda\Phi(y)$, whenever $x , y \in X$ and $0 \leq\lambda\leq1$.

A function $\Phi: X \rightarrow(-\infty, +\infty]$ is said to be lower-semi-continuous on X [14] if $\liminf_{y \rightarrow x}\Phi(y)\geq\Phi(x)$, for any $x \in X$.

Given a proper convex function Φ on X and a point $x \in X$, we denote by $\partial\Phi(x)$ the set of all $x^{*} \in X^{*}$ such that $\Phi(x)\leq\Phi(y)+ (x - y, x^{*})$, for any $y \in X$. Such element $x^{*}$ is called the subgradient of Φ at x, and $\partial\Phi(x)$ is called the subdifferential of Φ at x [14].

For easy reference of the reader, Theorems 1.1-1.5 mentioned in Section 1.1 are stated as follows.

Theorem 1.1

[1]

Let X be a real Banach space with a strictly convex dual space $X^{*}$. Let $J_{r}: X\rightarrow X^{*}$ be a duality mapping on X and there exists a function $\eta: X \rightarrow[0,+\infty)$ such that for all $u, v \in X$,

$$ \|J_{r}u - J_{r}v\| \leq\eta(u-v). $$

(1.14)

Let $A, C_{1}:X\rightarrow2^{X}$ be accretive mappings such that

• (i) either both A and $C_{1}$ satisfy the following condition (1.15), or $D(A)\subset D(C_{1})$ and $C_{1}$ satisfies the condition (1.15):

  $$ \left \{ \textstyle\begin{array}{l} \textit{for }u\in D(A)\textit{ and }v \in Au, \textit{there exists a constant }C(a,f) \textit{ such that} \\ (v - f, J_{r}(u - a)) \geq C(a,f); \end{array}\displaystyle \right . $$

  (1.15)

• (ii) $A+C_{1}$ is m-accretive and boundedly-inversely-compact.

Let $C_{2}:X\rightarrow X$ be a bounded continuous mapping such that, for any $y\in X$, there is a constant $C(y)$ satisfying $(C_{2}(u+y),J_{r}u)\ge-C(y)$ for any $u\in X$. Then the following results hold:

• (a) $\overline{[R(A)+R(C_{1})]}\subset\overline {R(A+C_{1}+C_{2})}$;

• (b) $\operatorname{int}[R(A)+R(C_{1})]\subset \operatorname{int} R(A+C_{1}+C_{2})$.

Theorem 1.2

[5]

Let $T: X \rightarrow X^{*}$ be a bounded and pseudo-monotone operator, and K be a closed and convex subset of X. Suppose that Φ is a lower-semi-continuous and convex function defined on K, which is not always +∞, such that $\Phi(v) \in(-\infty, +\infty]$, for any $v \in K$. Suppose there exists $v_{0} \in K$ such that $\Phi(v_{0})<+\infty$, and

$$\frac{(v - v_{0}, Tv) + \Phi(v)}{\|v\|} \rightarrow\infty, $$

as $\|v\|\rightarrow\infty$, $v \in K$. Then there exists $u \in K$ such that $(u - v,Tu) \leq\Phi(v)- \Phi(u)$, for all $v \in K$.

Theorem 1.3

[8]

Let X be a real reflexive Banach space with $X^{*}$ being its dual space. Let C be a non-empty closed convex subset of X. Assume that

• (i) the mapping $A: C \rightarrow2^{X^{*}}$ is a maximal monotone operator;

• (ii) the mapping $B: C \rightarrow X^{*}$ is pseudo-monotone, bounded, and demi-continuous;

• (iii) if the subset C is unbounded, then the operator B is A-coercive with respect to the fixed element $b \in X^{*}$, i.e., there exist an element $u_{0} \in C \cap D(A)$ and a number $r>0$ such that

  $$ (u - u_{0}, Bu)> (u - u_{0}, b), $$

  (1.16)

  for all $u \in C$ with $\|u\| > r$.

Then the equation $b \in Au + Bu$ has a solution.

Theorem 1.4

[10]

Let X be a smooth Banach space, $A : D(A) \subset X \rightarrow2^{X}$ be an m-accretive mapping, and $S: D(S)\subset X \rightarrow X$ be continuous and strongly accretive with $\overline{D(A)}\subset D(S)$. Then, for any $z \in X$, the equation $z \in Sx + \lambda Ax$ has a unique solution $x_{\lambda}$, for any $\lambda> 0$.

Theorem 1.5

[12]

Let X be a real reflexive Banach space with both X and $X^{*}$ being strictly convex. Let $J : X \rightarrow X^{*}$ be the normalized duality mapping on X. Let A and B be two maximal monotone operators in X. If there exist $0 \leq k < 1$ and $C_{1}, C_{2} > 0$ such that

$$ \bigl(a, J^{-1}(B_{t} v)\bigr) \geq- k \|B_{t} v \|^{2}-C_{1} \|B_{t} v\| - C_{2}, $$

(1.17)

for any $v \in D(A)$, $a \in Av$ and $t > 0$ ($B_{t}$ is the Yosida approximation of B), then $R(A) + R(B) \simeq R(A+B)$.

The following results will be needed in subsequent discussion.

Lemma 1.1

[14]

If A and B are maximal monotone operators in X such that $(\operatorname{int} D(A))\cap D(B) \neq\emptyset$, then $A+B$ is maximal monotone.

Lemma 1.2

[14]

If $\Phi: X \rightarrow R$ is proper, convex, and lower-semi-continuous, then ∂Φ is maximal monotone.

Lemma 1.3

[14]

If $B: X \rightarrow2^{X^{*}}$ is everywhere defined, monotone, and hemi-continuous, then B is maximal monotone.

Theorem 1.6

[8]

Assume that X is a real reflexive Banach space and the following conditions hold:

• (H1) The linear operator $L: D(L) \subseteq X \rightarrow X^{*}$ is maximal monotone in X.

• (H2) The operator $A: X \rightarrow2^{X^{*}}$ is monotone.

• (H3) The functional $\varphi: X \rightarrow(-\infty, +\infty]$ is convex, lower-semi-continuous, and $\varphi\neq +\infty$.

• (H4) One of the following conditions is satisfied:

  • (H4.1) $A : X \rightarrow X^{*}$ is single-valued and hemi-continuous;

  • (H4.2) A is maximal monotone and $\operatorname{int} D(A) \cap D(\partial \varphi)\neq\emptyset$;

  • (H4.3) A is maximal monotone and $D(A) \cap \operatorname{int} D(\partial \varphi)\neq\emptyset$.

• (H5) The sum $L+A+\partial\varphi: X \rightarrow2^{X^{*}}$ is coercive with respect to 0, i.e., there exist $r > 0$ and $u_{0} \in D(L) \cap D(A) \cap D(\partial\varphi)$ such that

  $$\bigl(u - u_{0}, u^{*}\bigr) > 0, $$

  for all $(u,u^{*}) \in L + A +\partial\varphi$ with $\|u\|> r$.

• (H6) $D(L) \cap D(A+\partial\varphi)\neq\emptyset$.

Then the equation

$$0 \in Lu + Au + \partial\varphi(u), \quad u \in X, $$

has a solution.

Definition 1.1

For $1 < p < +\infty$ and $1 < q < +\infty$, we use Y to denote the product of two spaces $W^{1,p}(\Omega)$ and $W^{1,q}(\Omega)$, i.e., $Y = W^{1,p}(\Omega)\times W^{1,q}(\Omega)= \{(u,v): u \in W^{1,p}(\Omega), v \in W^{1,q}(\Omega)\}$. The dual space of Y will be denoted by $Y^{*}$. Also, Y will be endowed with the norm

$$\bigl\Vert (u,v)\bigr\Vert _{Y} = \sqrt{\|u \|_{1,p,\Omega}^{2}+\|v\|_{1,q,\Omega}^{2}},\quad (u,v) \in Y, $$

where $\|\cdot\|_{1,p,\Omega}$ and $\|\cdot\|_{1,q,\Omega}$ denote the norm in $W^{1,p}(\Omega)$ and $W^{1,q}(\Omega)$, respectively.

Definition 1.2

[15]

For $1 < p < +\infty$, let $L^{p}(0,T ; X)$ denote the space of all X-valued strongly measurable functions $x(t)$ defined a.e. on $(0, T)$ such that $\|x(t)\|^{p}_{X}$ is Lebesgue integrable over $(0,T)$. It is well known that $L^{p}(0,T; X)$ is a Banach space with the norm defined by $\|x\|_{L^{p}(0,T ; X)} = (\int_{0}^{T} \|x(t)\|_{X} ^{p}\,dt )^{\frac{1}{p}}$. If X is reflexive, then $L^{p}(0,T ; X)$ is reflexive, and its dual space coincides with $L^{p'}(0,T ; X^{*})$, where $\frac{1}{p}+ \frac{1}{p'}= 1$. Moreover, $L^{p}(0,T ; X)$ is reflexive in the case when X is reflexive, and $L^{p}(0,T ; X)$ is strictly (uniformly) convex in the case when X is strictly (uniformly) convex.

Definition 1.3

For $1 < p < +\infty$ and $1 < q < +\infty$, we use Z to denote the product of two spaces $L^{p}(0,T; W^{1,p}(\Omega))$ and $L^{q}(0,T; W^{1,q}(\Omega))$, i.e., $Z = L^{p}(0,T; W^{1,p}(\Omega))\times L^{q}(0,T; W^{1,q}(\Omega)) = \{(u,v): u \in L^{p}(0,T; W^{1,p}(\Omega)), v \in L^{q}(0,T; W^{1,q}(\Omega))\}$. The dual space of Z is denoted by $Z^{*}$. Also, Z will be endowed with the norm

$$\bigl\Vert (u,v)\bigr\Vert _{Z} = \sqrt{\|u \|_{L^{p}(0,T; W^{1,p}(\Omega))}^{2}+\|v\|_{L^{q}(0,T; W^{1,q}(\Omega))}^{2}},\quad (u,v) \in Z. $$

Discussion of $(p,q)$-Laplacian elliptic system (1.10)

Throughout the paper, we shall assume that

$$\begin{aligned}& \frac{1}{p}+\frac{1}{p'} = 1,\qquad \frac{1}{q}+ \frac{1}{q'} = 1,\qquad N \geq1, \qquad \frac{2N}{N+1} < p < +\infty, \qquad \frac{2N}{N+1} < q < +\infty, \\& \frac{2N}{N+1} < r \leq\min\bigl\{ p,p'\bigr\} < +\infty,\qquad \frac{2N}{N+1} < s \leq\min\bigl\{ q,q'\bigr\} < +\infty. \end{aligned}$$

In (1.10)-(1.12), Ω is a bounded conical domain of the Euclidean space $\mathbb{R}^{N}$ with its boundary $\Gamma\in C^{1}$, ϑ denotes the exterior normal derivative to Γ, and $\varepsilon_{1}$ and $\varepsilon_{2}$ are non-negative constants. Let $\varphi: \Gamma\times\mathbb{R}\rightarrow\mathbb{R}$ be a given function such that, for each $x\in\Gamma$, $\varphi_{x}= \varphi(x,\cdot): \mathbb{R}\rightarrow\mathbb{R}$ is a proper, convex, and lower-semi-continuous function with $\varphi_{x}(0) = 0$. Let $\beta_{x}$ be the subdifferential of $\varphi_{x}$, i.e., $\beta_{x} \equiv \partial \varphi_{x}$. Suppose $0 \in\beta_{x}(0)$ and for each $t \in\mathbb{R}$, the function $x \in\Gamma\rightarrow(I+\lambda\beta_{x})^{-1}(t)\in \mathbb{R}$ is measurable for $\lambda> 0$.

In (1.10)-(1.12), suppose that $g_{i}:\Omega\times \mathbb{R}^{N+1} \rightarrow\mathbb{R}$ are given functions ($i = 1,2$) satisfying the following conditions, which can be found in [8, 16]:

• (a) Carathéodory’s conditions.

  For $i=1,2$, $x \rightarrow g_{i}(x,r)$ is measurable on Ω, for all $r \in\mathbb{R}^{N+1}$; $r \rightarrow g_{i}(x,r)$ is continuous on $\mathbb{R}^{N+1}$, for almost all $x\in\Omega$.

• (b) Growth condition.

  $$\begin{aligned}& g_{1}(x,s_{1},\ldots,s_{N+1})\leq h_{1}(x)+ k_{1} \sum_{i = 1}^{N+1} |s_{i}|^{p-1}, \\& g_{2}(x,s_{1},\ldots,s_{N+1})\leq h_{2}(x)+ k_{2} \sum_{i = 1}^{N+1} |s_{i}|^{q-1}, \end{aligned}$$

  where $(s_{1}, s_{2}, \ldots, s_{N+1})\in\mathbb{R}^{N+1}$, $h_{1}(x)\in L^{p}(\Omega)$, $h_{2}(x)\in L^{q}(\Omega)$ and $k_{i}$ are positive constants, $i = 1,2$.

• (c) Monotone condition.

  For $i=1,2$, $g_{i}(x,r_{1},\ldots,r_{N+1})$ is monotone with respect to $r_{1}$, i.e.,

  $$\bigl[g_{i}(x,s_{1},\ldots,s_{N+1}) - g_{i}(x,t_{1},\ldots,t_{N+1})\bigr](s_{1} - t_{1}) \geq 0, $$

  for all $x \in\Omega$ and $(s_{1},\ldots,s_{N+1}), (t_{1},\ldots,t_{N+1})\in\mathbb{R}^{N+1}$.

• (d) For $i=1,2$, $g_{i}(x, 0, \ldots, 0)\equiv0$, for $x \in \Omega$ and $(0, \ldots, 0) \in\mathbb{R}^{N+1}$.

Specific to system (1.10)

In (1.10), $f_{1}$, $f_{2}$, $C_{1}$, and $C_{2}$ are given functions with $f_{1}(x)\in L^{p'}(\Omega)$, $f_{2}(x)\in L^{q'}(\Omega)$, $0 \leq C_{1}(x) \in L^{p}(\Omega)$ and $0 \leq C_{2}(x) \in L^{q}(\Omega)$.

Lemma 2.1

[2, 16]

Define the operators $B_{1}: W^{1,p}(\Omega) \rightarrow(W^{1,p}(\Omega))^{*}$ and $B_{2}: W^{1,q}(\Omega) \rightarrow(W^{1,q}(\Omega))^{*}$ by

$$(w,B_{1}u) = \int_{\Omega}\bigl\langle \bigl(C_{1}(x)+|\nabla u|^{2}\bigr)^{\frac {p-2}{2}}\nabla u, \nabla w \bigr\rangle \,dx +\varepsilon_{1}\int_{\Omega}|u|^{r-2}uw \,dx, \quad u,w \in W^{1,p}(\Omega) $$

and

$$(w,B_{2}v) = \int_{\Omega}\bigl\langle \bigl(C_{2}(x)+|\nabla v|^{2}\bigr)^{\frac {q-2}{2}}\nabla v, \nabla w \bigr\rangle \,dx +\varepsilon_{2} \int_{\Omega}|v|^{s-2}vw \,dx,\quad v,w \in W^{1,q}(\Omega). $$

Then $B_{i}$, $i=1,2$, is everywhere defined, strictly monotone, hemi-continuous, and coercive. Moreover, it is noted from Lemma  1.3 that $B_{i}$, $i=1,2$, is maximal monotone. (Here $\langle\cdot,\cdot\rangle$ and $|\cdot|$ denote the Euclidean inner-product and Euclidean norm in $\mathbb{R}^{N}$, respectively.)

Definition 2.1

Define $A: Y \rightarrow Y^{*}$ by $A(u,v)=(B_{1} u, B_{2} v)$, for $(u,v)\in Y$.

Proposition 2.1

The mapping $A: Y \rightarrow Y^{*}$ is everywhere defined, monotone, and hemi-continuous.

Proof

Step 1. A is everywhere defined.

In fact, for any $(u, v), (w_{1}, w_{2}) \in Y$, we have $|((w_{1}, w_{2}), A(u, v))| = |((w_{1},w_{2}), (B_{1}u,B_{2}v))| \leq|(w_{1}, B_{1} u)|+|(w_{2}, B_{2} v)|$. Since $B_{1}$ and $B_{2}$ are everywhere defined, A is everywhere defined.

Step 2. A is monotone.

To show this, let $(w^{(1)}_{1}, w_{2}^{(1)} ), (w^{(2)}_{1}, w_{2}^{(2)} ) \in Y$, then

$$\begin{aligned}& \bigl(\bigl(w^{(1)}_{1}, w_{2}^{(1)} \bigr)-\bigl(w^{(2)}_{1}, w_{2}^{(2)} \bigr), A\bigl(w^{(1)}_{1}, w_{2}^{(1)} \bigr)- A\bigl(w^{(2)}_{1}, w_{2}^{(2)} \bigr) \bigr) \\& \quad = \bigl(w^{(1)}_{1}-w^{(2)}_{1}, B_{1}w^{(1)}_{1} - B_{1} w_{1}^{(2)} \bigr) + \bigl(w^{(1)}_{2} - w_{2}^{(2)}, B_{2}w^{(1)}_{2} -B_{2} w_{2}^{(2)} \bigr). \end{aligned}$$

Since both $B_{1}$ and $B_{2}$ are monotone, A is monotone.

Step 3. A is hemi-continuous.

It suffices to show that for any $(u_{1} , u_{2}), (v_{1}, v_{2}), (w_{1}, w_{2}) \in Y$ and $k \in[0,1]$,

$$\bigl((w_{1},w_{2}), A\bigl((u_{1}, u_{2})+k (v_{1}, v_{2})\bigr)- A(u_{1}, u_{2}) \bigr) \rightarrow0, $$

as $k \rightarrow0$. In fact, notice that both $B_{1}$ and $B_{2}$ are hemi-continuous, A is also hemi-continuous. □

Lemma 2.2

[2]

The mapping $\Phi_{1}: W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

$$\Phi_{1}(u) = \int_{\Gamma}\varphi_{x} \bigl(u|_{\Gamma}(x)\bigr)\,d\Gamma(x),\quad u \in W^{1,p}(\Omega) $$

is proper, convex, and lower-semi-continuous on $W^{1,p}(\Omega)$. The subdifferential $\partial\Phi_{1}$ of $\Phi_{1}$ is maximal monotone in view of Lemma  1.2. Moreover,

$$\bigl(w_{1}, \partial\Phi_{1}(u)\bigr) = \int _{\Gamma} \beta_{x}\bigl(u|_{\Gamma}(x) \bigr)w_{1}|_{\Gamma}(x) \,d\Gamma(x), \quad u, w_{1} \in W^{1,p}(\Omega). $$

The mapping $\Phi_{2}: W^{1,q}(\Omega)\rightarrow\mathbb{R}$ defined by

$$\Phi_{2}(v) = \int_{\Gamma}\varphi_{x} \bigl(v|_{\Gamma}(x)\bigr)\,d\Gamma(x),\quad v \in W^{1,q}(\Omega) $$

is proper, convex, and lower-semi-continuous on $W^{1,q}(\Omega)$. The subdifferential $\partial\Phi_{2}$ of $\Phi_{2}$ is maximal monotone in view of Lemma  1.2. Moreover,

$$\bigl(w_{2}, \partial\Phi_{2}(v)\bigr) = \int _{\Gamma} \beta_{x}\bigl(v|_{\Gamma}(x) \bigr)w_{2}|_{\Gamma}(x) \,d\Gamma(x), \quad v, w_{2} \in W^{1,q}(\Omega). $$

Proposition 2.2

The mapping $\Phi: Y \rightarrow\mathbb{R}$ defined by

$$\Phi(u,v) = \int_{\Gamma}\varphi_{x} \bigl(u|_{\Gamma}(x)\bigr)\,d\Gamma(x)+\int_{\Gamma}\varphi_{x}\bigl(v|_{\Gamma}(x)\bigr)\,d\Gamma(x),\quad (u,v)\in Y $$

is proper, convex, and lower-semi-continuous on Y. The subdifferential ∂Φ of Φ is maximal monotone in view of Lemma  1.2. Moreover,

$$\partial\Phi(u,v) = \bigl(\partial\Phi_{1}(u),\partial \Phi_{2}(v)\bigr). $$

Proof

Since $\Phi_{1}$ and $\Phi_{2}$ are proper, it is not difficult to find that Φ is also proper.

For $0 \leq\lambda\leq1$ and $(u_{1},v_{1}), (u_{2}, v_{2})\in Y$, we find

$$\begin{aligned}& \Phi\bigl((1-\lambda) (u_{1},v_{1})+\lambda(u_{2},v_{2}) \bigr) \\& \quad = \Phi \bigl((1-\lambda)u_{1}+\lambda u_{2}, (1- \lambda)v_{1}+\lambda v_{2}\bigr) \\& \quad = \int_{\Gamma}\varphi_{x}\bigl((1- \lambda)u_{1}|_{\Gamma}(x)+\lambda u_{2}|_{\Gamma}(x) \bigr)\,d\Gamma(x) +\int_{\Gamma}\varphi_{x}\bigl((1- \lambda )v_{1}|_{\Gamma}(x)+\lambda v_{2}|_{\Gamma}(x) \bigr)\,d\Gamma(x) \\& \quad \leq(1-\lambda)\int_{\Gamma}\varphi_{x} \bigl(u_{1}|_{\Gamma}(x)\bigr)\,d\Gamma(x)+\lambda\int _{\Gamma}\varphi_{x}\bigl(u_{2}|_{\Gamma}(x) \bigr)\,d\Gamma(x) \\& \qquad {} +(1-\lambda)\int_{\Gamma}\varphi_{x} \bigl(v_{1}|_{\Gamma}(x)\bigr)\,d\Gamma(x)+\lambda\int _{\Gamma}\varphi_{x}\bigl(v_{2}|_{\Gamma}(x) \bigr)\,d\Gamma(x) \\& \quad = (1-\lambda)\Phi(u_{1},v_{1})+ \lambda \Phi(u_{2}, v_{2}), \end{aligned}$$

which implies that Φ is convex.

For $(w,z) \in Y$, since $\Phi_{1}$ and $\Phi_{2}$ are lower-semi-continuous, we have

$$\begin{aligned} \liminf_{(u,v) \rightarrow (w,z)}\Phi(u,v) =& \liminf_{(u,v) \rightarrow (w,z)} \biggl[\int_{\Gamma}\varphi_{x}\bigl(u|_{\Gamma}(x) \bigr)\,d\Gamma(x)+\int_{\Gamma}\varphi_{x} \bigl(v|_{\Gamma}(x)\bigr)\,d\Gamma(x) \biggr] \\ \geq& \int_{\Gamma}\varphi_{x}\bigl(w|_{\Gamma}(x) \bigr)\,d\Gamma(x)+\int_{\Gamma}\varphi_{x} \bigl(z|_{\Gamma}(x)\bigr)\,d\Gamma(x) = \Phi(w,z), \end{aligned}$$

which implies that Φ is lower-semi-continuous.

For $(u,v), (w,z) \in Y$, in view of the definition of the subdifferential, we get

$$\Phi_{1}(u) + \Phi_{2}(v) \leq\Phi_{1}(w) + \bigl(u - w, \partial\Phi _{1}(u) \bigr) + \Phi_{2}(z)+ \bigl(v - z, \partial\Phi_{2}(v) \bigr). $$

Then

$$\begin{aligned}& \int_{\Gamma}\varphi_{x}\bigl(u|_{\Gamma}(x) \bigr)\,d\Gamma(x) +\int_{\Gamma}\varphi_{x} \bigl(v|_{\Gamma}(x)\bigr)\,d\Gamma(x) \\& \quad \leq\int_{\Gamma}\varphi_{x} \bigl(w|_{\Gamma}(x)\bigr)\,d\Gamma(x)+\bigl(u - w, \partial \Phi_{1}(u)\bigr) +\int_{\Gamma}\varphi_{x} \bigl(z|_{\Gamma}(x)\bigr)\,d\Gamma(x)+\bigl(v-z, \partial \Phi_{2}(v)\bigr), \end{aligned}$$

which implies that

$$\Phi(u,v) \leq \Phi(w,z) + \bigl((u,v) - (w,z), \bigl(\partial \Phi_{1}(u), \partial\Phi_{2}(v)\bigr) \bigr). $$

Thus,

$$\partial\Phi(u,v) = \bigl(\partial\Phi_{1}(u), \partial \Phi_{2}(v)\bigr). $$

This completes the proof. □

Lemma 2.3

[16]

Define $G_{1} : W^{1,p}(\Omega) \rightarrow(W^{1,p}(\Omega))^{*}$ by

$$(w, G_{1}u) = \int_{\Omega}g_{1}(x,u, \nabla u)w \,dx,\quad u,w \in W^{1,p}(\Omega). $$

Then $G_{1}$ is everywhere defined, monotone, and hemi-continuous on $W^{1,p}(\Omega)$.

Define $G_{2} : W^{1,q}(\Omega) \rightarrow(W^{1,q}(\Omega))^{*}$ by

$$(w, G_{2}v) = \int_{\Omega}g_{2}(x,v,\nabla v)w \,dx,\quad v,w \in W^{1,q}(\Omega). $$

Then $G_{2}$ is everywhere defined, monotone, and hemi-continuous on $W^{1,q}(\Omega)$.

Proposition 2.3

Define $G : Y \rightarrow Y^{*}$ by

$$\bigl((w,z),G(u,v)\bigr) = (w, G_{1}u)+(z, G_{2}v),\quad (u,v) \in Y. $$

Then G is everywhere defined, monotone, and hemi-continuous on Y. Moreover, G is maximal monotone.

Proof

The result follows from Lemma 2.3 and the definition of G. □

Theorem 2.1

For $f_{1}(x)\in L^{p'}(\Omega)$ and $f_{2}(x)\in L^{q'}(\Omega)$, the nonlinear $(p,q)$-Laplacian elliptic system (1.10) has a unique solution in Y.

Proof

Define $T : Y \rightarrow Y^{*}$ by

$$\begin{aligned} \bigl((w_{1} , w_{2}),T(u,v)\bigr) =& \bigl((w_{1} , w_{2}),A(u,v)\bigr)+\bigl((w_{1},w_{2}), G(u,v) \bigr) \\ &{}- \int_{\Omega}f_{1} w_{1} \,dx - \int_{\Omega}f_{2} w_{2} \,dx, \end{aligned}$$

for $(u, v), (w_{1}, w_{2})\in Y$. From Propositions 2.1 and 2.3, $T : Y \rightarrow Y^{*}$ is everywhere defined, monotone, hemi-continuous, and then it is maximal monotone.

Combining with Lemma 1.1 and Proposition 2.2, we know that $T+\partial\Phi$ is maximal monotone.

Next, we shall show that

$$\lim_{\|(u,v)\|_{Y} \rightarrow+\infty} \frac{((u,v), T(u,v)+ \partial\Phi(u,v))}{\|(u,v)\|_{Y} } = + \infty. $$

Noting that $\partial\Phi(0,0) = (0,0)$, $G(0,0) = (0,0)$ and G is monotone, we have

$$\begin{aligned} \frac{((u, v), T(u,v)+ \partial\Phi(u,v))}{\|(u,v)\|_{Y}} \geq& \frac{(u, B_{1}u)+(v, B_{2} v)}{\|(u,v)\|_{Y}}- \frac{\int_{\Omega}f_{1}u \,dx + \int_{\Omega} f_{2}v \,dx }{\|(u,v)\|_{Y}} \\ \geq& \frac{(u, B_{1}u)+(v, B_{2} v)}{\|(u,v)\|_{Y}}- \|f_{1}\|_{L^{p'}(\Omega)} - \|f_{2}\|_{L^{q'}(\Omega)}. \end{aligned}$$

Let $\|(u,v)\|_{Y} \rightarrow+\infty$, then $\|u\|_{1,p,\Omega} \rightarrow+ \infty$ or $\|v\|_{1,q,\Omega} \rightarrow+ \infty$.

Case 1. If $\|u\|_{1,p,\Omega} \rightarrow+ \infty$ and $\|v\|_{1,q,\Omega} \leq \mathrm{const}$, then

$$\begin{aligned} \frac{((u, v), T(u,v)+ \partial\Phi(u,v))}{\|(u,v)\|_{Y}} \geq& \frac{(u, B_{1}u)}{\|(u,v)\|_{Y}} - \|f_{1} \|_{L^{p'}(\Omega)} - \|f_{2}\|_{L^{q'}(\Omega)} \\ = & \frac{(u, B_{1}u)}{\|u\|_{1,p,\Omega}}\times \frac{1}{\sqrt{1+\frac{\|v\|_{1,q,\Omega}^{2}}{\|u\|_{1,p,\Omega}^{2}}}}- \|f_{1} \|_{L^{p'}(\Omega)} - \|f_{2}\|_{L^{q'}(\Omega)} \\ \rightarrow& +\infty, \end{aligned}$$

as $\|(u,v)\|_{Y} \rightarrow+\infty$, since $B_{1}$ is coercive.

Case 2. If $\|u\|_{1,p,\Omega}\leq \mathrm{const}$ and $\|v\|_{1,q,\Omega} \rightarrow+ \infty$, then the proof is similar to that of Case 1.

Case 3. If $\|u\|_{1,p,\Omega} \rightarrow+ \infty$ and $\|v\|_{1,q,\Omega} \rightarrow+ \infty$, then we split the discussion into the following cases:

• (i) Suppose $\frac{\|u\|_{1,p,\Omega}}{\|v\|_{1,q,\Omega}} \rightarrow+ \infty$. In this case,

  $$\begin{aligned} \frac{((u, v), T(u,v)+ \partial\Phi(u,v))}{\|(u,v)\|_{Y}} \geq& \frac{(u, B_{1}u)}{\sqrt{\|u\|_{1,p,\Omega}^{2}+\|v\|_{1,q,\Omega}^{2}}} - \|f_{1} \|_{L^{p'}(\Omega)} - \|f_{2}\|_{L^{q'}(\Omega)} \\ = & \frac{(u, B_{1}u)}{\|u\|_{1,p,\Omega} \sqrt{1+\frac{\|v\|_{1,q,\Omega}^{2}}{\|u\|_{1,p,\Omega}^{2}}}}- \|f_{1}\|_{L^{p'}(\Omega)} - \|f_{2}\|_{L^{q'}(\Omega)} \\ \rightarrow& +\infty, \end{aligned}$$

  since $B_{1}$ is coercive.

• (ii) Suppose $\frac{\|v\|_{1,q,\Omega}}{\|u\|_{1,p,\Omega}} \rightarrow+ \infty$. Similar to case (i), the result follows.

• (iii) Suppose $\frac{\|u\|_{1,p,\Omega}}{\|v\|_{1,q,\Omega}} \rightarrow \mathrm{const} \neq0$. In this case,

  $$\begin{aligned} \frac{((u, v), T(u,v)+\partial\Phi(u,v))}{\|(u,v)\|_{Y}} \geq&\frac{(u, B_{1}u)}{\|u\|_{1,p,\Omega} \sqrt{1+\frac{\|v\|_{1,q,\Omega}^{2}}{\|u\|_{1,p,\Omega}^{2}}}}- \|f_{1}\|_{L^{p'}(\Omega)} - \|f_{2}\|_{L^{q'}(\Omega)} \\ \rightarrow& +\infty, \end{aligned}$$

  since $B_{1}$ is coercive.

Therefore, for $r > 0$, there always exists $(0,0) \in D(T) \cap D(\partial\Phi)$ such that

$$\bigl((u,v), T(u,v) + \partial\Phi(u,v)\bigr) > 0, $$

for all $(u,v) \in Y$ with $\|(u,v)\|_{Y} > r$.

Then, in view of Theorem 1.6, the equation

$$ (0,0) = T(u,v) + \partial\Phi(u,v) $$

(2.1)

has a solution in Y, which is denoted by $(u,v)$. From the strict monotonicity of $B_{1}$ and $B_{2}$, $(u,v)$ is unique. Next, we shall show that this $(u,v)$ is the solution of (1.10).

For $(\varphi,\varphi)\in C_{0}^{\infty}(\Omega)\times C_{0}^{\infty}(\Omega)$, using (2.1) we find

$$\begin{aligned}& \int_{\Omega} \bigl\langle \bigl(C_{1}(x)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u, \nabla\varphi \bigr\rangle \,dx + \varepsilon_{1} \int_{\Omega}|u|^{r-2} u \varphi \,dx+ \int_{\Omega} g_{1}(x,u,\nabla u)\varphi \,dx \\& \qquad {} + \int_{\Omega} \bigl\langle \bigl(C_{2}(x)+| \nabla v|^{2}\bigr)^{\frac {q-2}{2}}\nabla v, \nabla\varphi \bigr\rangle \,dx+ \varepsilon_{2} \int_{\Omega }|v|^{s-2} v \varphi \,dx+ \int_{\Omega} g_{2}(x,v,\nabla v)\varphi \,dx \\& \qquad {}- \int_{\Omega}f_{1} \varphi \,dx - \int _{\Omega}f_{2} \varphi \,dx +\bigl(\varphi, \partial \Phi_{1}(u)\bigr) +\bigl(\varphi, \partial\Phi_{2}(v)\bigr) \\& \quad = -\int_{\Omega}\operatorname{div} \bigl[ \bigl(C_{1}(x)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr]\varphi \,dx+ \varepsilon_{1} \int_{\Omega}|u|^{r-2} u \varphi \,dx+\int_{\Omega} g_{1}(x,u,\nabla u) \varphi \,dx \\& \qquad {}-\int_{\Omega}\operatorname{div} \bigl[ \bigl(C_{2}(x)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr]\varphi \,dx + \varepsilon_{2} \int_{\Omega}|u|^{s-2} u \varphi \,dx+\int_{\Omega} g_{2}(x,v,\nabla v) \varphi \,dx \\& \qquad {}- \int_{\Omega}f_{1} \varphi \,dx - \int _{\Omega}f_{2} \varphi \,dx = 0. \end{aligned}$$

From the property of generalized function, we have

$$\begin{aligned}& -\operatorname{div} \bigl[\bigl(C_{1}(x)+|\nabla u|^{2} \bigr)^{\frac {p-2}{2}}\nabla u \bigr]+ \varepsilon_{1} |u|^{r-2} u + g_{1}(x,u,\nabla u) -\operatorname{div} \bigl[ \bigl(C_{2}(x)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr] \\& \quad {}+ \varepsilon_{2} |v|^{s-2} v + g_{2}(x,v, \nabla v) = f_{1}(x) + f_{2}(x). \end{aligned}$$

(2.2)

Using Green’s formula and (2.1), we have, for $(w_{1},0) \in Y$,

$$\begin{aligned} 0 =& \int_{\Omega} \bigl\langle \bigl(C_{1}(x)+| \nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u, \nabla w_{1} \bigr\rangle \,dx + \varepsilon_{1} \int_{\Omega}|u|^{r-2} u w_{1} \,dx+ \int_{\Omega} g_{1}(x,u, \nabla u)w_{1} \,dx\,dt \\ &{}- \int_{\Omega}f_{1} w_{1} \,dx + \bigl(w_{1}, \partial\Phi_{1}(u)\bigr) \\ = & -\int_{\Omega}\operatorname{div} \bigl[ \bigl(C_{1}(x)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr]w_{1} \,dx+\int_{\Gamma} \bigl\langle \vartheta, \bigl(C_{1}(x)+|\nabla u |^{2}\bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle w_{1} \,d\Gamma(x) \\ &{} + \varepsilon_{1} \int_{\Omega}|u|^{r-2} u w_{1} \,dx+\int_{\Omega} g_{1}(x,u,\nabla u) w_{1} \,dx - \int_{\Omega}f_{1} w_{1} \,dx + \int_{\Gamma}\beta_{x}(u) w_{1} \,d\Gamma(x). \end{aligned}$$

Then

$$\begin{aligned} 0 =& -\operatorname{div} \bigl[\bigl(C_{1}(x)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr]+ \bigl\langle \vartheta, \bigl(C_{1}(x)+|\nabla u |^{2}\bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle \\ &{} + \varepsilon_{1} |u|^{r-2} u + g_{1}(x,u, \nabla u) - f_{1}+\beta_{x}(u). \end{aligned}$$

(2.3)

Similarly, using Green’s formula and (2.1), we have, for $(0,w_{2}) \in Y$,

$$\begin{aligned} 0 =& \int_{\Omega} \bigl\langle \bigl(C_{2}(x)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v, \nabla w_{2} \bigr\rangle \,dx+ \varepsilon_{2} \int_{\Omega}|v|^{s-2} v w_{2} \,dx+ \int_{\Omega} g_{2}(x,v,\nabla v)w_{2} \,dx\,dt \\ &{} - \int_{\Omega}f_{2} w_{2} \,dx + \bigl(w_{2}, \partial\Phi_{2}(v)\bigr) \\ = & -\int_{\Omega}\operatorname{div} \bigl[ \bigl(C_{2}(x,t)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr]w_{2} \,dx+\int_{\Gamma} \bigl\langle \vartheta, \bigl(C_{2}(x)+|\nabla v |^{2}\bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle w_{2} \,d\Gamma(x) \\ &{} + \varepsilon_{2} \int_{\Omega}|v|^{s-2} v w_{2} \,dx+\int_{\Omega} g_{2}(x,v,\nabla v) w_{2} \,dx - \int_{\Omega}f_{2} w_{2} \,dx + \int_{\Gamma}\beta_{x}(v) w_{2} \,d\Gamma(x). \end{aligned}$$

Then

$$\begin{aligned} 0 =& -\operatorname{div} \bigl[\bigl(C_{2}(x)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr]+ \bigl\langle \vartheta, \bigl(C_{2}(x)+|\nabla v |^{2}\bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle \\ &{} + \varepsilon_{2} |v|^{s-2} v + g_{2}(x,v, \nabla v) - f_{2}+\beta_{x}(v). \end{aligned}$$

(2.4)

Since $(u,v)$ satisfies (2.2), by using (2.3) and (2.4) we have

$$- \bigl\langle \vartheta, \bigl(C_{1}(x)+|\nabla u |^{2} \bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle - \bigl\langle \vartheta, \bigl(C_{2}(x)+|\nabla v |^{2}\bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle \in\beta_{x}(u)+\beta_{x}(v), \quad \mbox{a.e. }x \in\Gamma. $$

Thus, $(u,v)$ is the solution of (1.10). This completes the proof. □

Discussion of $(p,q)$-Laplacian parabolic system (1.11)

We recall that Ω, Γ, ϑ, $\varepsilon_{1}$, $\varepsilon_{2}$, $\beta_{x}$, $g_{1}$, and $g_{2}$ satisfy the conditions stated at the beginning of Section 2.

Specific to system (1.11)

In (1.11), T is a constant, $f_{1}$, $f_{2}$, $C_{1}$, and $C_{2}$ are given functions with $f_{1}(x)\in (L^{p}(0, T; W^{1,p}(\Omega)))^{*}$, $f_{2}(x)\in(L^{q}(0,T; W^{1,q}(\Omega)))^{*}$, $0 \leq C_{1}(x,t)\in L^{p}(\Omega\times(0,T))$, and $0 \leq C_{2}(x,t)\in L^{q}(\Omega\times(0,T))$.

Lemma 3.1

[9]

Define the operators $\widetilde{B}_{1}: L^{p}(0, T; W^{1,p}(\Omega)) \rightarrow(L^{p}(0, T; W^{1,p}(\Omega)))^{*}$ and $\widetilde{B}_{2}: L^{q}(0, T; W^{1,q}(\Omega)) \rightarrow(L^{q}(0, T; W^{1,q}(\Omega)))^{*}$ by

$$\begin{aligned}& (w,\widetilde{B}_{1}u) = \int_{0}^{T} \int_{\Omega}\bigl\langle \bigl(C_{1}(x,t)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u, \nabla w \bigr\rangle \,dx\,dt + \varepsilon_{1}\int_{0}^{T}\int _{\Omega}|u|^{r-2}uw \,dx\,dt, \\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\ \ \ \, \,u,w \in L^{p}\bigl(0, T; W^{1,p}(\Omega)\bigr) \end{aligned}$$

and

$$\begin{aligned}& (w,\widetilde{B}_{2}v) = \int_{0}^{T} \int_{\Omega}\bigl\langle \bigl(C_{2}(x,t)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v, \nabla w \bigr\rangle \,dx\,dt + \varepsilon_{2} \int_{0}^{T}\int _{\Omega}|v|^{s-2}vw \,dx\,dt, \\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\ \ \ \, v,w \in L^{q}\bigl(0, T; W^{1,q}(\Omega)\bigr). \end{aligned}$$

Then $\widetilde{B}_{i}$, $i=1,2$, is everywhere defined, strictly monotone, hemi-continuous, and coercive. Moreover, $\widetilde{B}_{i}$, $i=1,2$, is maximal monotone.

Definition 3.1

Define $\widetilde{A}: Z \rightarrow Z^{*}$ by $\widetilde{A}(u,v)=(\widetilde{B}_{1} u, \widetilde{B}_{2} v)$, for $(u,v)\in Z$.

Proposition 3.1

[9]

The mapping $\widetilde{A}: Z \rightarrow Z^{*}$ is everywhere defined, maximal monotone, hemi-continuous, and coercive.

Lemma 3.2

[7, 9]

The mapping $\widetilde{\Phi}_{1}: L^{p}(0, T; W^{1,p}(\Omega))\rightarrow\mathbb{R}$ defined by

$$\widetilde{\Phi}_{1}(u) = \int_{0}^{T} \int_{\Gamma}\varphi_{x}\bigl(u|_{\Gamma}(x,t) \bigr)\,d\Gamma(x)\,dt, \quad u \in L^{p}\bigl(0, T; W^{1,p}( \Omega)\bigr) $$

is proper, convex, and lower-semi-continuous on $L^{p}(0, T; W^{1,p}(\Omega))$. The subdifferential $\partial \widetilde{\Phi}_{1}$ of $\widetilde{\Phi}_{1}$ is maximal monotone in view of Lemma  1.2. Moreover,

$$\bigl(w_{1}, \partial\widetilde{\Phi}_{1}(u)\bigr) = \int _{0}^{T} \int_{\Gamma} \beta_{x}\bigl(u|_{\Gamma}(x,t)\bigr)w_{1}|_{\Gamma}(x,t) \,d\Gamma(x)\,dt, \quad u, w_{1} \in L^{p}\bigl(0, T; W^{1,p}(\Omega)\bigr). $$

The mapping $\widetilde{\Phi}_{2}: L^{q}(0, T; W^{1,q}(\Omega))\rightarrow\mathbb{R}$ defined by

$$\widetilde{\Phi}_{2}(v) = \int_{0}^{T} \int_{\Gamma}\varphi_{x}\bigl(v|_{\Gamma}(x) \bigr)\,d\Gamma(x)\,dt , \quad v \in L^{q}\bigl(0, T; W^{1,q}( \Omega)\bigr) $$

is proper, convex, and lower-semi-continuous on $L^{q}(0, T; W^{1,q}(\Omega))$. The subdifferential $\partial\widetilde{\Phi}_{2}$ of $\widetilde{\Phi}_{2}$ is maximal monotone in view of Lemma  1.2. Moreover,

$$\bigl(w_{2}, \partial\widetilde{\Phi}_{2}(v)\bigr) = \int _{0}^{T} \int_{\Gamma}\beta_{x}\bigl(v|_{\Gamma}(x,t)\bigr)w_{2}|_{\Gamma}(x,t) \, d \Gamma(x)\,dt,\quad v, w_{2} \in L^{q}\bigl(0, T; W^{1,q}(\Omega)\bigr). $$

Proposition 3.2

The mapping $\widetilde{\Phi}: Z \rightarrow Z^{*}$ defined by

$$\widetilde{\Phi}(u,v) = \int_{0}^{T}\int _{\Gamma}\varphi_{x}\bigl(u|_{\Gamma}(x,t)\bigr) \,d\Gamma(x)\,dt +\int_{0}^{T}\int _{\Gamma}\varphi_{x}\bigl(v|_{\Gamma}(x,t)\bigr) \,d\Gamma(x)\,dt,\quad (u,v)\in Z $$

is proper, convex, and lower-semi-continuous on Z. The subdifferential ∂Φ̃ of Φ̃ is maximal monotone in view of Lemma  1.2. Moreover,

$$\partial\widetilde{\Phi}(u,v) = \bigl(\partial\widetilde{\Phi} _{1}(u),\partial\widetilde{\Phi}_{2}(v)\bigr). $$

Proof

The proof is similar to that of Proposition 2.2. □

Lemma 3.3

Define $\widetilde{G}_{1} : L^{p}(0, T; W^{1,p}(\Omega)) \rightarrow(L^{p}(0, T; W^{1,p}(\Omega)))^{*}$ by

$$(w, \widetilde{G}_{1}u) = \int_{0}^{T} \int_{\Omega}g_{1}(x,u,\nabla u)w \,dx\,dt, \quad u,w \in L^{p}\bigl(0, T; W^{1,p}(\Omega)\bigr). $$

Then $\widetilde{G}_{1}$ is everywhere defined, monotone, and hemi-continuous on $L^{p}(0, T; W^{1,p}(\Omega))$.

Define $\widetilde{G}_{2} : L^{q}(0, T; W^{1,q}(\Omega)) \rightarrow (L^{q}(0, T; W^{1,q}(\Omega)))^{*}$ by

$$(w, \widetilde{G}_{2}v) = \int_{0}^{T} \int_{\Omega}g_{2}(x,v,\nabla v)w \,dx\,dt, \quad v,w \in L^{q}\bigl(0, T; W^{1,q}(\Omega)\bigr). $$

Then $\widetilde{G}_{2}$ is everywhere defined, monotone and hemi-continuous on $L^{q}(0, T; W^{1,q}(\Omega))$.

Proof

The proof is similar to that of Lemma 2.3. □

Proposition 3.3

Define $\widetilde{G} : Z \rightarrow Z^{*}$ by

$$\bigl((w,z),\widetilde{G}(u,v)\bigr) = (w, \widetilde{G}_{1}u)+(z, \widetilde{G}_{2}v), \quad (u,v) \in Z. $$

Then G̃ is maximal monotone.

Proof

The result follows from Lemma 3.3 and the definition of G̃. □

Lemma 3.4

[7, 9]

Define $S_{1}: D(S_{1}) = \{u \in L^{p}(0, T; W^{1,p}(\Omega)): \frac{\partial u}{\partial t} \in (L^{p}(0, T; W^{1,p}(\Omega)))^{*}, u(x,0) = u(x,T)\}\rightarrow(L^{p}(0, T; W^{1,p}(\Omega)))^{*}$ by

$$S_{1} u(x,t) = \frac{\partial u}{\partial t}. $$

Then $S_{1}$ is a linear maximal monotone operator possessing a dense domain in $L^{p}(0, T; W^{1,p}(\Omega))$.

Define $S_{2}: D(S_{2}) = \{v \in L^{q}(0, T; W^{1,q}(\Omega)): \frac{\partial v}{\partial t} \in(L^{q}(0, T; W^{1,q}(\Omega)))^{*}, v(x,0) = v(x,T) \}\rightarrow(L^{q}(0, T; W^{1,q}(\Omega)))^{*}$ by

$$S_{2} v(x,t) = \frac{\partial v}{\partial t}. $$

Then $S_{2}$ is a linear maximal monotone operator possessing a dense domain in $L^{q}(0, T; W^{1,q}(\Omega))$.

Proposition 3.4

[9]

Define $S : Z \rightarrow Z^{*}$ by

$$\bigl((w,z),S(u,v)\bigr) = (w, S_{1}u)+(z, S_{2}v),\quad (u,v) \in D(S). $$

Then S is linear maximal monotone.

Theorem 3.1

For $(f_{1}(x), f_{2}(x))\in Z^{*}$, the nonlinear $(p,q)$-Laplacian parabolic system (1.11) has a unique solution in Z.

Proof

Define $\widetilde{T} : Z \rightarrow Z^{*}$ by

$$\begin{aligned} \bigl((w_{1} , w_{2}),\widetilde{T}(u,v)\bigr) =& \bigl((w_{1} , w_{2}),\widetilde{A}(u,v)\bigr)+ \bigl((w_{1},w_{2}), \widetilde{G}(u,v)\bigr) \\ &{}- \int _{0}^{T}\int_{\Omega}f_{1} w_{1} \,dx\,dt - \int_{0}^{T} \int_{\Omega}f_{2} w_{2} \,dx\,dt, \end{aligned}$$

for $(u, v), (w_{1}, w_{2})\in Z$. From Propositions 3.1 and 3.3, $\widetilde{T} : Z \rightarrow Z^{*}$ is everywhere defined, monotone, hemi-continuous, and then it is maximal monotone.

Using Lemma 1.1 and Propositions 3.1 and 3.2, we know that $\widetilde{T}+\partial\widetilde{\Phi}$ is maximal monotone.

Next, we shall show that

$$\lim_{\|(u,v)\|_{Z} \rightarrow+\infty} \frac{((u,v), S(u,v)+\widetilde{T}(u,v)+ \partial\widetilde{\Phi}(u,v))}{\|(u,v)\|_{Z} } = + \infty. $$

Noting that $\partial\widetilde{\Phi}(0,0) = (0,0)$, $\widetilde{G}(0,0) = 0$ and $S(0,0) = (0,0)$, we have

$$\begin{aligned}& \frac{((u, v), S(u,v)+\widetilde{T}(u,v)+ \partial \widetilde{\Phi}(u,v))}{\|(u,v)\|_{Z}} \\& \quad \geq\frac{(u, \widetilde{B}_{1}u)+(v, \widetilde{B}_{2} v)}{\|(u,v)\|_{Z}}- \frac{\int_{0}^{T}\int_{\Omega}f_{1}u \,dx\,dt + \int_{0}^{T}\int_{\Omega} f_{2}v \,dx\,dt }{\|(u,v)\|_{Z}} \\& \quad \geq\frac{(u, \widetilde{B}_{1}u)+(v, \widetilde{B}_{2} v)}{\|(u,v)\|_{Z}} - \|f_{1}\|_{(L^{p}(0, T; W^{1,p}(\Omega)))^{*}} - \|f_{2}\|_{(L^{q}(0, T; W^{1,q}(\Omega)))^{*}}. \end{aligned}$$

Let $\|(u,v)\|_{Z} \rightarrow+\infty$, then $\|u\|_{L^{p}(0, T; W^{1,p}(\Omega))} \rightarrow+ \infty$ or $\|v\|_{L^{q}(0, T; W^{1,q}(\Omega))} \rightarrow+ \infty$.

Case 1. If $\|u\|_{L^{p}(0, T; W^{1,p}(\Omega))} \rightarrow+ \infty$ and $\|v\|_{L^{q}(0, T; W^{1,q}(\Omega))} \leq \mathrm{const}$, then

$$\begin{aligned}& \frac{((u, v), \widetilde{T}(u,v)+ \partial\widetilde{\Phi}(u,v))}{\|(u,v)\|_{Z}} \\& \quad \geq\frac{(u, \widetilde{B}_{1}u)}{\|(u,v)\|_{Z}} - \|f_{1}\|_{(L^{p}(0, T; W^{1,p}(\Omega)))^{*}} - \|f_{2}\|_{(L^{q}(0, T; W^{1,q}(\Omega)))^{*}} \\& \quad = \frac{(u, \widetilde{B}_{1}u)}{\|u\|_{L^{p}(0, T; W^{1,p}(\Omega))}}\times\frac{1}{\sqrt{1+\frac{\|v\|_{L^{q}(0, T; W^{1,q}(\Omega))}^{2}}{\|u\|_{L^{p}(0, T; W^{1,p}(\Omega))}^{2}}}} - \|f_{1} \|_{(L^{p}(0, T; W^{1,p}(\Omega)))^{*}} - \|f_{2}\|_{(L^{q}(0, T; W^{1,q}(\Omega)))^{*}} \\ & \quad \rightarrow+\infty, \end{aligned}$$

as $\|(u,v)\|_{Z} \rightarrow+\infty$, since $\widetilde{B}_{1}$ is coercive.

Case 2. If $\|u\|_{L^{p}(0, T; W^{1,p}(\Omega))}\leq \mathrm{const}$ and $\|v\|_{L^{q}(0, T; W^{1,q}(\Omega))} \rightarrow+ \infty$, then the proof is similar to that of Case 1.

Case 3. If $\|u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}} \rightarrow+ \infty$ and $\|v\|_{{L^{q}(0, T; W^{1,q}(\Omega))}} \rightarrow+ \infty$, then we split the discussion into the following cases:

• (i) Suppose $\frac{\|u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}}}{\|v\| _{{L^{q}(0, T; W^{1,q}(\Omega))}}} \rightarrow+ \infty$. In this case,

  $$\begin{aligned}& \frac{((u, v), \widetilde{T}(u,v)+ \partial\widetilde{\Phi}(u,v))}{\|(u,v)\|_{Z}} \\& \quad \geq\frac{(u, \widetilde{B}_{1}u)}{\sqrt{\|u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}}^{2}+\|v\|_{{L^{q}(0, T; W^{1,q}(\Omega))}}^{2}}} \\& \qquad {}- \|f_{1}\|_{({L^{p}(0, T; W^{1,p}(\Omega))})^{*}} - \|f_{2}\|_{({L^{q}(0, T; W^{1,q}(\Omega))})^{*}} \\& \quad = \frac{(u, \widetilde{B}_{1}u)}{\|u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}} \sqrt{1+\frac{\|v\|_{{L^{q}(0, T; W^{1,q}(\Omega))}}^{2}}{\|u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}}^{2}}}} \\& \qquad {}- \|f_{1}\|_{({L^{p}(0, T; W^{1,p}(\Omega))})^{*}} - \|f_{2}\|_{({L^{q}(0, T; W^{1,q}(\Omega))})^{*}} \\& \quad \rightarrow +\infty, \end{aligned}$$

  since $\widetilde{B}_{1}$ is coercive.

• (ii) Suppose $\frac{\|v\|_{{L^{q}(0, T; W^{1,q}(\Omega))}}}{\| u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}}} \rightarrow+ \infty$. Similar to case (i), the result follows.

• (iii) Suppose $\frac{\|u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}}}{\|v\|_{{L^{q}(0, T; W^{1,q}(\Omega))}}} \rightarrow \mathrm{const} \neq0$. In this case,

  $$\begin{aligned}& \frac{((u, v), \widetilde{T}(u,v)+\partial \widetilde{\Phi}(u,v))}{\|(u,v)\|_{Z}} \\& \quad \geq \frac{(u, \widetilde{B}_{1}u)}{\|u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}} \sqrt{1+\frac{\|v\|_{{L^{q}(0, T; W^{1,q}(\Omega))}}^{2}}{\|u\|_{{L^{p}(0, T; W^{1,p}(\Omega))}}^{2}}}} \\& \qquad {}- \|f_{1}\|_{({L^{p}(0, T; W^{1,p}(\Omega))})^{*}} - \|f_{2}\|_{({L^{q}(0, T; W^{1,q}(\Omega))})^{*}} \\& \quad \rightarrow+\infty, \end{aligned}$$

  since $\widetilde{B}_{1}$ is coercive.

Therefore, for $r > 0$, there always exists $(0,0) \in D(S) \cap D(\widetilde{T}) \cap D(\partial\widetilde{\Phi})$ such that

$$\bigl((u,v), S(u,v)+\widetilde{T}(u,v) + \partial\widetilde{\Phi}(u,v)\bigr) > 0, $$

for all $(u,v) \in Z$ with $\|(u,v)\|_{Z} > r$.

Then, in view of Theorem 1.6, the equation

$$ (0,0) = S(u,v)+\widetilde{T}(u,v) + \partial\widetilde{\Phi}(u,v) $$

(3.1)

has a solution in Z, which is denoted by $(u,v)$. From the strict monotonicity of $\widetilde{B}_{1}$ and $\widetilde{B}_{2}$, this $(u,v)$ is unique. Next, we shall show that this $(u,v)$ is the solution of (1.11).

For $(\varphi,\varphi)\in C_{0}^{\infty}(0,T; \Omega)\times C_{0}^{\infty}(0,T; \Omega)$, using (3.1) we find

$$\begin{aligned}& \int_{0}^{T} \int_{\Omega} \frac{\partial u}{\partial t} \varphi \,dx\,dt+\int_{0}^{T} \int_{\Omega} \bigl\langle \bigl(C_{1}(x,t)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u, \nabla\varphi \bigr\rangle \,dx \,dt \\& \qquad {}+ \varepsilon_{1} \int_{0}^{T} \int _{\Omega}|u|^{r-2} u \varphi \,dx\,dt+ \int_{0}^{T} \int_{\Omega}g_{1}(x,u, \nabla u)\varphi \,dx\,dt \\& \qquad {}+ \int_{0}^{T} \int _{\Omega}\frac{\partial v}{\partial t} \varphi \,dx\,dt + \int _{0}^{T} \int_{\Omega} \bigl\langle \bigl(C_{2}(x,t)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v, \nabla\varphi \bigr\rangle \,dx\,dt \\& \qquad {}+ \varepsilon_{2} \int_{0}^{T} \int_{\Omega}|v|^{s-2} v \varphi \,dx\,dt+ \int _{0}^{T}\int_{\Omega} g_{2}(x,v,\nabla v)\varphi \,dx\,dt - \int_{0}^{T} \int_{\Omega}f_{1} \varphi \,dx\,dt \\& \qquad {}- \int _{0}^{T} \int_{\Omega}f_{2} \varphi \,dx\,dt+ \bigl(\varphi, \partial \widetilde{\Phi}_{1}(u)\bigr) + \bigl(\varphi, \partial\widetilde{\Phi}_{2}(v)\bigr) \\& \quad = \int_{0}^{T} \int_{\Omega} \frac{\partial u}{\partial t} \varphi \,dx\,dt -\int_{0}^{T} \int_{\Omega}\operatorname{div} \bigl[\bigl(C_{1}(x,t)+| \nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr]\varphi \,dx\,dt \\& \qquad {}+ \varepsilon_{1} \int_{0}^{T} \int _{\Omega}|u|^{r-2} u \varphi \,dx\,dt+\int_{0}^{T} \int_{\Omega} g_{1}(x,u,\nabla u) \varphi \,dx\,dt \\& \qquad {}+ \int_{0}^{T} \int_{\Omega}\frac{\partial v}{\partial t} \varphi \,dx\,dt -\int _{0}^{T} \int_{\Omega} \operatorname{div} \bigl[\bigl(C_{2}(x,t)+|\nabla v|^{2} \bigr)^{\frac{q-2}{2}}\nabla v \bigr]\varphi \,dx\,dt \\& \qquad {}+ \varepsilon_{2} \int_{0}^{T} \int_{\Omega}|v|^{s-2} v \varphi \,dx\,dt+\int _{0}^{T}\int_{\Omega} g_{2}(x,v,\nabla v) \varphi \,dx\,dt \\& \qquad {}- \int_{0}^{T} \int_{\Omega}f_{1} \varphi \,dx\,dt - \int _{0}^{T} \int_{\Omega}f_{2} \varphi \,dx\,dt = 0. \end{aligned}$$

From the property of generalized function, we have

$$\begin{aligned}& \frac{\partial u(x,t)}{\partial t}-\operatorname{div} \bigl[\bigl(C_{1}(x,t)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr]+ \varepsilon_{1} |u|^{r-2} u + g_{1}(x,u,\nabla u)+\frac {\partial v(x,t)}{\partial t} \\& \quad {}- \operatorname{div} \bigl[\bigl(C_{2}(x,t)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr]+ \varepsilon_{2} |v|^{s-2} v + g_{2}(x,v,\nabla v) = f_{1}(x,t) + f_{2}(x,t). \end{aligned}$$

(3.2)

Using Green’s formula and (3.1), we have, for $(w_{1},0) \in Z$,

$$\begin{aligned} 0 =&\int_{0}^{T} \int_{\Omega} \frac{\partial u}{\partial t} w_{1} \,dx\,dt+\int_{0}^{T} \int_{\Omega} \bigl\langle \bigl(C_{1}(x,t)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u, \nabla w_{1} \bigr\rangle \,dx\,dt \\ &{}+ \varepsilon_{1} \int_{0}^{T} \int_{\Omega}|u|^{r-2} u w_{1} \,dx\,dt + \int_{0}^{T}\int_{\Omega} g_{1}(x,u,\nabla u)w_{1} \,dx\,dt \\ &{}- \int _{0}^{T}\int_{\Omega}f_{1} w_{1} \,dx\,dt + \bigl(w_{1}, \partial\widetilde{ \Phi}_{1}(u)\bigr) \\ =& \int_{0}^{T} \int_{\Omega} \frac{\partial u}{\partial t} w_{1} \,dx\,dt -\int_{0}^{T} \int_{\Omega}\operatorname{div} \bigl[\bigl(C_{1}(x,t)+| \nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr]w_{1} \,dx \,dt \\ &{} +\int_{0}^{T}\int_{\Gamma} \bigl\langle \vartheta, \bigl(C_{1}(x,t)+|\nabla u |^{2} \bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle w_{1} \,d\Gamma(x)\,dt + \varepsilon_{1} \int_{0}^{T}\int _{\Omega}|u|^{r-2} u w_{1} \,dx\,dt \\ &{} +\int_{0}^{T}\int_{\Omega} g_{1}(x,u,\nabla u) w_{1} \,dx\,dt- \int _{0}^{T}\int_{\Omega}f_{1} w_{1} \,dx\,dt+\int_{0}^{T}\int _{\Gamma }\beta_{x}(u) w_{1} \,d\Gamma(x) \,dt. \end{aligned}$$

Then

$$\begin{aligned}& \frac{\partial u}{\partial t}-\operatorname{div} \bigl[\bigl(C_{1}(x,t)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr]+ \bigl\langle \vartheta, \bigl(C_{1}(x,t)+|\nabla u |^{2}\bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle \\& \quad {} + \varepsilon_{1} |u|^{r-2} u + g_{1}(x,u,\nabla u)- f_{1} +\beta_{x}(u) = 0. \end{aligned}$$

(3.3)

Similarly, using Green’s formula and (3.1), we have, for $(0,w_{2}) \in Z$,

$$\begin{aligned} 0 =& \int_{0}^{T} \int_{\Omega} \frac{\partial v}{\partial t} w_{2} \,dx\,dt + \int_{0}^{T} \int_{\Omega} \bigl\langle \bigl(C_{2}(x,t)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v, \nabla w_{2} \bigr\rangle \,dx\,dt \\ &{} + \varepsilon_{2} \int_{0}^{T}\int _{\Omega}|v|^{s-2} v w_{2} \,dx\,dt+ \int _{0}^{T}\int_{\Omega} g_{2}(x,v,\nabla v)w_{2} \,dx\,dt \\ &{}- \int_{0}^{T} \int_{\Omega}f_{2} w_{2} \,dx\,dt + \bigl(w_{2}, \partial\widetilde{\Phi }_{2}(v)\bigr) \\ = & \int_{0}^{T} \int_{\Omega} \frac{\partial v}{\partial t} w_{2} \,dx\,dt -\int_{0}^{T} \int_{\Omega}\operatorname{div} \bigl[\bigl(C_{2}(x,t)+| \nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr]w_{2} \,dx \,dt \\ &{} +\int_{0}^{T}\int_{\Gamma} \bigl\langle \vartheta, \bigl(C_{2}(x,t)+|\nabla v |^{2} \bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle w_{2} \,d\Gamma(x)\,dt + \varepsilon_{2} \int_{0}^{T}\int _{\Omega}|v|^{s-2} v w_{2} \,dx\,dt \\ &{}+\int_{0}^{T}\int_{\Omega} g_{2}(x,v,\nabla v) w_{2} \,dx\,dt - \int _{0}^{T}\int_{\Omega}f_{2} w_{2} \,dx+ \int_{0}^{T}\int _{\Gamma}\beta_{x}(v) w_{2} \,d\Gamma(x)\,dt. \end{aligned}$$

Then

$$\begin{aligned}& \frac{\partial v}{\partial t}-\operatorname{div} \bigl[\bigl(C_{2}(x,t)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr]+ \bigl\langle \vartheta, \bigl(C_{2}(x,t)+|\nabla v |^{2}\bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle \\& \quad {} + \varepsilon_{2} |v|^{s-2} v + g_{2}(x,v, \nabla v)- f_{2} +\beta_{x}(v) = 0. \end{aligned}$$

(3.4)

Since $(u,v)$ satisfies (3.2), by using (3.3) and (3.4) we have

$$\begin{aligned}& - \bigl\langle \vartheta, \bigl(C_{1}(x,t)+|\nabla u |^{2} \bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle - \bigl\langle \vartheta, \bigl(C_{2}(x,t)+|\nabla v |^{2}\bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle \\& \quad \in \beta_{x}(u)+\beta_{x}(v),\quad (x,t)\in \Gamma\times(0,T). \end{aligned}$$

Hence, $(u,v)$ is the solution of (1.11). This completes the proof. □

Discussion of $(p,q)$-Laplacian integro-differential system (1.12)

We recall that Ω, Γ, ϑ, $\varepsilon_{1}$, $\varepsilon_{2}$, $\beta_{x}$, $g_{1}$, and $g_{2}$ satisfy the conditions stated at the beginning of Section 2.

Specific to system (1.12)

In (1.12), T, $a_{1}$, and $a_{2}$ are non-negative constants, $f_{1}$, $f_{2}$, $C_{1}$, and $C_{2}$ are given functions with $f_{1}(x)\in(L^{p}(0, T; W^{1,p}(\Omega)))^{*}$, $f_{2}(x)\in(L^{q}(0,T; W^{1,q}(\Omega)))^{*}$, $0 \leq C_{1}(x,t)\in L^{p}(\Omega\times(0,T))$ and $0 \leq C_{2}(x,t)\in L^{q}(\Omega\times (0,T))$.

Lemma 4.1

[9]

Define $\widetilde{S}_{1}: D(\widetilde{S}_{1}) = \{u \in L^{p}(0, T; W^{1,p}(\Omega)): \frac{\partial u}{\partial t} \in(L^{p}(0, T; W^{1,p}(\Omega)))^{*}, u(x,0)= u(x,T)\} \rightarrow(L^{p}(0, T; W^{1,p}(\Omega)))^{*}$ by

$$\widetilde{S}_{1} u(x,t) = \frac{\partial u}{\partial t}+a_{1} \frac{\partial}{\partial t}\int_{\Omega}u \,dx. $$

Then $\widetilde{S}_{1}$ is a linear maximal monotone operator possessing a dense domain in $L^{p}(0, T; W^{1,p}(\Omega))$.

Define $\widetilde{S}_{2}: D(\widetilde{S}_{2}) = \{v \in L^{q}(0, T; W^{1,q}(\Omega)): \frac{\partial v}{\partial t} \in(L^{q}(0, T; W^{1,q}(\Omega)))^{*}, v(x,0) = v(x,T)\}\rightarrow(L^{q}(0, T; W^{1,q}(\Omega)))^{*}$ by

$$\widetilde{S}_{2} v(x,t) = \frac{\partial v}{\partial t}+a_{2} \frac{\partial}{\partial t}\int_{\Omega}v \,dx. $$

Then $\widetilde{S}_{2}$ is a linear maximal monotone operator possessing a dense domain in $L^{q}(0, T; W^{1,q}(\Omega))$.

Proposition 4.1

Define $\widetilde{S} : Z \rightarrow Z^{*}$ by

$$\bigl((w,z),\widetilde{S}(u,v)\bigr) = (w, \widetilde{S}_{1}u)+(z, \widetilde{S}_{2}v), \quad (u,v) \in Z. $$

Then S̃ is linear maximal monotone.

Theorem 4.1

For $(f_{1}(x,t), f_{2}(x,t))\in Z^{*}$, the nonlinear $(p,q)$-Laplacian integro-differential system (1.12) has a unique solution in Z.

Proof

Define $\widetilde{T}, \partial\widetilde{\Phi}: Z \rightarrow Z^{*}$ as in Theorem 3.1 and Proposition 3.2, respectively.

Since $\widetilde{S}(0,0) = (0,0)$, similar to the proof of Theorem 3.1, for $r> 0$, there always exists $(0,0) \in D(\widetilde{S}) \cap D(\widetilde{T}) \cap D(\partial \widetilde{\Phi})$ such that

$$\bigl((u,v), \widetilde{S}(u,v)+\widetilde{T}(u,v) + \partial\widetilde { \Phi}(u,v)\bigr) > 0, $$

for all $(u,v) \in Z$ with $\|(u,v)\|_{Z} > r$.

In view of Theorem 1.6, the equation

$$ (0,0) = \widetilde{S}(u,v)+\widetilde{T}(u,v) + \partial\widetilde{\Phi}(u,v) $$

(4.1)

has a unique solution in Z, which is denoted by $(u,v)$. As in the proof of Theorem 3.1, this $(u,v)$ is unique. Next, we shall show that this $(u,v)$ is the solution of (1.12).

For $(\varphi,\varphi)\in C_{0}^{\infty}(0,T; \Omega)\times C_{0}^{\infty}(0,T; \Omega)$, using (4.1) we have

$$\begin{aligned}& \int_{0}^{T}\int_{\Omega} \frac{\partial u}{\partial t}\varphi \,dx\,dt + \int_{0}^{T} \int_{\Omega} \biggl(a_{1} \frac{\partial}{\partial t}\int _{\Omega}u \,dx \biggr) \,dx\,dt \\& \qquad {}+ \int_{0}^{T} \int_{\Omega} \bigl\langle \bigl(C_{1}(x,t)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u, \nabla\varphi \bigr\rangle \,dx \,dt \\& \qquad {} + \varepsilon_{1} \int_{0}^{T} \int_{\Omega}|u|^{r-2} u \varphi \,dx\,dt + \int _{0}^{T} \int_{\Omega}g_{1}(x,u, \nabla u)\varphi \,dx\,dt+ \int_{0}^{T}\int _{\Omega}\frac{\partial v}{\partial t}\varphi \,dx\,dt \\& \qquad {} + \int_{0}^{T} \int _{\Omega} \biggl(a_{2} \frac{\partial }{\partial t}\int _{\Omega}v \,dx \biggr) \,dx\,dt + \int_{0}^{T} \int_{\Omega} \bigl\langle \bigl(C_{2}(x,t)+|\nabla v|^{2}\bigr)^{\frac {q-2}{2}}\nabla v, \nabla\varphi \bigr\rangle \,dx \,dt \\& \qquad {} + \varepsilon_{2} \int_{0}^{T} \int_{\Omega}|v|^{s-2} v \varphi \,dx\,dt + \int _{0}^{T}\int_{\Omega} g_{2}(x,v,\nabla v)\varphi \,dx\,dt - \int_{0}^{T} \int_{\Omega}f_{1} \varphi \,dx\,dt \\& \qquad {} - \int_{0}^{T} \int _{\Omega}f_{2} \varphi \,dx\,dt +\bigl(\varphi, \partial \widetilde{\Phi}_{1}(u)\bigr) +\bigl(\varphi, \partial\widetilde{ \Phi}_{2}(v)\bigr) \\& \quad = \int_{0}^{T}\int_{\Omega} \frac{\partial u}{\partial t}\varphi \,dx\,dt + \int_{0}^{T} \int_{\Omega} \biggl(a_{1} \frac{\partial}{\partial t}\int _{\Omega}u \,dx \biggr) \,dx\,dt \\& \qquad {}-\int_{0}^{T} \int_{\Omega}\operatorname{div} \bigl[\bigl(C_{1}(x,t)+| \nabla u|^{2}\bigr)^{\frac {p-2}{2}}\nabla u \bigr]\varphi \,dx\,dt \\& \qquad {} + \varepsilon_{1} \int_{0}^{T} \int_{\Omega}|u|^{r-2} u \varphi \,dx\,dt+\int _{0}^{T} \int_{\Omega} g_{1}(x,u,\nabla u) \varphi \,dx\,dt + \int_{0}^{T} \int_{\Omega}\frac{\partial v}{\partial t}\varphi \,dx\,dt \\& \qquad {} + \int_{0}^{T} \int _{\Omega} \biggl(a_{2} \frac{\partial }{\partial t}\int _{\Omega}v \,dx \biggr) \,dx\,dt -\int_{0}^{T} \int_{\Omega}\operatorname{div} \bigl[\bigl(C_{2}(x,t)+| \nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr]\varphi \,dx\,dt \\& \qquad {} + \varepsilon_{2} \int_{0}^{T} \int_{\Omega}|v|^{s-2} v \varphi \,dx\,dt+\int _{0}^{T}\int_{\Omega} g_{2}(x,v,\nabla v) \varphi \,dx\,dt \\& \qquad {}- \int_{0}^{T} \int_{\Omega}f_{1} \varphi \,dx\,dt - \int _{0}^{T} \int_{\Omega}f_{2} \varphi \,dx\,dt = 0. \end{aligned}$$

From the property of generalized function, we get

$$\begin{aligned}& \frac{\partial u(x,t)}{\partial t}-\operatorname{div} \bigl[\bigl(C_{1}(x,t)+|\nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr]+ \varepsilon_{1} |u|^{r-2} u + g_{1}(x,u,\nabla u)+a_{1} \frac{\partial}{\partial t}\int_{\Omega}u \,dx \\& \qquad {} + \frac{\partial v(x,t)}{\partial t}-\operatorname{div} \bigl[\bigl(C_{2}(x,t)+| \nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr] + \varepsilon_{2} |v|^{s-2} v + g_{2}(x,v,\nabla v) +a_{2}\frac{\partial}{\partial t}\int_{\Omega}v \,dx \\& \quad =f_{1}(x,t) + f_{2}(x,t). \end{aligned}$$

(4.2)

Using Green’s formula and (4.1), we have, for $(w_{1},0) \in Z$,

$$\begin{aligned} \begin{aligned} 0 ={}& \int_{0}^{T}\int_{\Omega} \frac{\partial u}{\partial t}w_{1} \,dx\,dt + \int_{0}^{T} \int_{\Omega} \biggl(a_{1} \frac{\partial}{\partial t}\int _{\Omega}u \,dx \biggr)w_{1} \,dx\,dt \\ &{} + \int_{0}^{T}\int_{\Omega} \bigl\langle \bigl(C_{1}(x,t)+|\nabla u|^{2} \bigr)^{\frac{p-2}{2}}\nabla u, \nabla w_{1} \bigr\rangle \,dx\,dt+ \varepsilon_{1} \int_{0}^{T}\int _{\Omega}|u|^{r-2} u w_{1} \,dx\,dt \\ &{} +\int_{0}^{T}\int_{\Omega} g_{1}(x,u,\nabla u)w_{1} \,dx\,dt- \int _{0}^{T}\int_{\Omega}f_{1} w_{1} \,dx\,dt + \bigl(w_{1}, \partial\widetilde{ \Phi}_{1}(u)\bigr) \\ = {}& \int_{0}^{T}\int_{\Omega} \frac{\partial u}{\partial t}w_{1} \,dx\,dt + \int_{0}^{T} \int_{\Omega} \biggl(a_{1} \frac{\partial}{\partial t}\int _{\Omega}u \,dx \biggr)w_{1} \,dx\,dt \\ &{} -\int_{0}^{T}\int_{\Omega} \operatorname{div} \bigl[\bigl(C_{1}(x,t)+|\nabla u|^{2} \bigr)^{\frac{p-2}{2}}\nabla u \bigr]w_{1} \,dx\,dt \\ &{}+\int_{0}^{T}\int_{\Gamma} \bigl\langle \vartheta, \bigl(C_{1}(x,t)+|\nabla u |^{2} \bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle w_{1} \,d\Gamma(x)\,dt+ \varepsilon_{1} \int_{0}^{T}\int _{\Omega}|u|^{r-2} u w_{1} \,dx\,dt \\ &{} +\int_{0}^{T}\int_{\Omega} g_{1}(x,u,\nabla u) w_{1} \,dx\,dt - \int _{0}^{T}\int_{\Omega}f_{1} w_{1} \,dx\,dt+ \int_{0}^{T}\int _{\Gamma}\beta_{x}(u) w_{1} \,d\Gamma(x) \,dt. \end{aligned} \end{aligned}$$

Then

$$\begin{aligned}& \frac{\partial u}{\partial t} + a_{1} \frac{\partial}{\partial t}\int _{\Omega}u \,dx -\operatorname{div} \bigl[\bigl(C_{1}(x,t)+| \nabla u|^{2}\bigr)^{\frac{p-2}{2}}\nabla u \bigr] \\& \quad {} + \bigl\langle \vartheta, \bigl(C_{1}(x,t)+|\nabla u |^{2}\bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle + \varepsilon_{1} |u|^{r-2} u + g_{1}(x,u,\nabla u)- f_{1} + \beta_{x}(u) = 0. \end{aligned}$$

(4.3)

Similarly, using Green’s formula and (4.1), we have, for $(0, w_{2}) \in Z$,

$$\begin{aligned} 0 =& \int_{0}^{T} \int_{\Omega} \frac{\partial v}{\partial t} w_{2} \,dx\,dt + \int_{0}^{T} \int_{\Omega} \biggl(a_{2} \frac{\partial}{\partial t}\int _{\Omega}v \,dx \biggr)w_{2} \,dx\,dt \\ &{} + \int_{0}^{T}\int_{\Omega} \bigl\langle \bigl(C_{2}(x,t)+|\nabla v|^{2} \bigr)^{\frac{q-2}{2}}\nabla v, \nabla w_{2} \bigr\rangle \,dx\,dt + \varepsilon_{2} \int_{0}^{T}\int _{\Omega}|v|^{s-2} v w_{2} \,dx\,dt \\ &{} +\int_{0}^{T}\int_{\Omega} g_{2}(x,v,\nabla v)w_{2} \,dx\,dt - \int_{0}^{T} \int_{\Omega}f_{2} w_{2} \,dx\,dt + \bigl(w_{2}, \partial\widetilde{\Phi }_{2}(v)\bigr) \\ = & \int_{0}^{T} \int_{\Omega} \frac{\partial v}{\partial t} w_{2} \,dx\,dt+ \int_{0}^{T} \int_{\Omega}\biggl(a_{2} \frac{\partial}{\partial t}\int _{\Omega}v \,dx\biggr)w_{2} \,dx\,dt \\ &{} -\int_{0}^{T}\int_{\Omega} \operatorname{div} \bigl[\bigl(C_{2}(x,t)+|\nabla v|^{2} \bigr)^{\frac{q-2}{2}}\nabla v \bigr]w_{2} \,dx\,dt \\ &{} +\int_{0}^{T}\int_{\Gamma} \bigl\langle \vartheta, \bigl(C_{2}(x,t)+|\nabla v |^{2} \bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle w_{2} \,d\Gamma(x)\,dt + \varepsilon_{2} \int_{0}^{T}\int _{\Omega}|v|^{s-2} v w_{2} \,dx\,dt \\ &{} +\int_{0}^{T}\int_{\Omega} g_{2}(x,v,\nabla v) w_{2} \,dx\,dt- \int_{0}^{T} \int_{\Omega}f_{2} w_{2} \,dx+ \int _{0}^{T}\int_{\Gamma} \beta_{x}(v) w_{2} \,d\Gamma(x)\,dt. \end{aligned}$$

Then

$$\begin{aligned}& \frac{\partial v}{\partial t}+ a_{2} \frac{\partial}{\partial t}\int_{\Omega}v \,dx-\operatorname{div} \bigl[\bigl(C_{2}(x,t)+|\nabla v|^{2}\bigr)^{\frac{q-2}{2}}\nabla v \bigr]+ \bigl\langle \vartheta, \bigl(C_{2}(x,t)+|\nabla v |^{2}\bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle \\& \quad {} + \varepsilon_{2} |v|^{s-2} v + g_{2}(x,v, \nabla v)- f_{2} +\beta_{x}(v) = 0. \end{aligned}$$

(4.4)

Since $(u,v)$ satisfies (4.2), using (4.3) and (4.4) we have

$$\begin{aligned}& - \bigl\langle \vartheta, \bigl(C_{1}(x,t)+|\nabla u |^{2} \bigr)^{\frac{p-2}{2}} \nabla u \bigr\rangle - \bigl\langle \vartheta, \bigl(C_{2}(x,t)+|\nabla v |^{2}\bigr)^{\frac{q-2}{2}} \nabla v \bigr\rangle \\& \quad \in\beta_{x}(u)+\beta_{x}(v),\quad (x,t)\in \Gamma\times(0,T). \end{aligned}$$

Thus, $(u,v)$ is the unique solution of (1.12). This completes the proof. □

Examples

In this section, we give some examples of the systems (1.10)-(1.12) discussed in this paper.

Example 5.1

We list two examples of (1.10) - the first system (5.1) is from [14] and the second system (5.2) is discussed in [17]. However, different methods have been employed:

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} -u'' + \varepsilon u = f(x), \\ - u' = 0. \end{array}\displaystyle \right . \end{aligned}$$

(5.1)

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} -\Delta u - \mu \Delta v = g(x,v), \quad x\in\Omega, \\ -\Delta v - \lambda \Delta u = f(x,u), \quad x\in\Omega, \\ u = v = 0,\quad x\in\Gamma. \end{array}\displaystyle \right . \end{aligned}$$

(5.2)

Example 5.2

The following system, which has been studied in [18], is an example of (1.12):

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u}{\partial t} - \Delta _{p} u + a \frac{\partial}{\partial t} \int_{\Omega}u \,dx = f(x,t), \quad (x,t)\in\Omega\times(0,T), \\ u(x,t) = 0,\quad x\in\Gamma, \\ u(x,0) = 0,\quad x \in\Omega. \end{array}\displaystyle \right . $$

(5.3)

Once again, different methods have been employed.

Example 5.3

If Ω reduces to a bounded interval $(a,b)$ in $\mathbb{R}^{1}$, examples of $\beta_{x}$ and $g_{i}$ ($i = 1,2$) can be found readily. For example, for $x \in\Gamma$, take $\varphi_{x} = \varphi(x,t)= t^{2}x^{2}$. Then $\varphi_{x} \equiv\varphi(x,\cdot): \mathbb{R} \rightarrow\mathbb{R}$ is a proper, convex, and lower-semi-continuous function. Further, $\beta_{x} = 2tx^{2}$. For $i =1,2$, take $g_{i}(x,t_{1},t_{2}): \Omega\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ as

$$g_{1}(x,t_{1},t_{2})= \left \{ \textstyle\begin{array}{l} \min\{a,x\}+ (2\max\{|t_{1}|^{p-1}, |t_{2}|^{p-1}\} -|t_{1}|^{p-1} )(\operatorname{sgn} t_{1}), \quad |t_{1}|\geq|t_{2}|, \\ \min\{a,x\}+ (2\min\{|t_{1}|^{p-1}, |t_{2}|^{p-1}\} -|t_{1}|^{p-1} )(\operatorname{sgn} t_{1}),\quad |t_{1}|\leq|t_{2}| \end{array}\displaystyle \right . $$

and

$$g_{2}(x,t_{1},t_{2})= \left \{ \textstyle\begin{array}{l} \min\{a,x\}+ (2\max\{|t_{1}|^{q-1}, |t_{2}|^{q-1}\} -|t_{1}|^{q-1} )(\operatorname{sgn}t_{1}), \quad |t_{1}|\geq|t_{2}|, \\ \min\{a,x\}+ (2\min\{|t_{1}|^{q-1}, |t_{2}|^{q-1}\} -|t_{1}|^{q-1} )(\operatorname{sgn} t_{1}), \quad |t_{1}|\leq|t_{2}|. \end{array}\displaystyle \right . $$

Then $g_{i}$ satisfies the assumptions (a)-(c). If, $a \equiv0$, then the assumption (d) is also satisfied.