An elliptic system with logarithmic nonlinearity

TL;DR

This paper investigates the existence of solutions for elliptic systems with logarithmic nonlinearities involving variable exponent Laplacians, using bifurcation theory and subsupersolution methods.

Contribution

It introduces a novel approach combining bifurcation theory and subsupersolution methods to handle singular elliptic systems with variable exponent operators.

Findings

Existence results for solutions to the elliptic system

Application of bifurcation theory to singular systems

Development of subsupersolution techniques for variable exponent operators

Abstract

In the present paper we study the existence of solutions for some classes of singular systems involving the p(x) and q(x) Laplacian operators. The approach is based on bifurcation theory and subsupersolution method for systems of quasilinear equations involving singular terms.

Equations

$$\left\{\begin{array}[]{ll}-\Delta_{p(x)}u=-\gamma\log v+\theta v^{\alpha(x)}&\ \mbox{in}\ \Omega,\\ -\Delta_{q(x)}v=-\gamma\log u+\theta u^{\beta(x)}&\ \mbox{in}\ \Omega,\\ u,v>0&\ \mbox{in}\ \Omega,\\ u=v=0&\ \mbox{on}\ \partial\Omega,\end{array}\right.$$

$$p^{\prime}(x)\leq p(x)^{\ast}\text{, }q^{\prime}(x)\leq q(x)^{\ast}\text{ \ and }\left\{\begin{array}[]{l}1<p^{-}\leq p^{+}<N\\ 1<q^{-}\leq q^{+}<N,\end{array}\right.$$

$$\begin{array}[]{l}s^{-}=\inf_{x\in\Omega}s(x)\text{ \ and \ }s^{+}=\sup_{x\in\Omega}s(x)\text{ for }s\in C(\overline{\Omega}).\end{array}$$

$$u(x),v(x)\ge cd(x)\text{ in }\Omega ,$$

$$\left\{\begin{array}[]{c}\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\nabla\phi\ dx=\int_{\Omega}(-\gamma\log v+\theta v^{\alpha(x)})\phi\ dx\\ \int_{\Omega}|\nabla v|^{q(x)-2}\nabla v\nabla\psi\ dx=\int_{\Omega}(-\gamma\log u+\theta u^{\beta(x)})\psi\ dx,\end{array}\right.$$

$$0<{\alpha }^{-}\le {\alpha }^{+}<q^{-}-1,0<{\beta }^{-}\le {\beta }^{+}<p^{-}-1$$

$${\alpha }^{-}>q^{+}-1,{\beta }^{-}>p^{+}-1,$$

$${\alpha }^{+}>q^{-}-1\text{and }{\beta }^{+}>p^{-}-1,$$

$$\left\{\begin{array}[]{l}\frac{\gamma}{\theta e}<\alpha^{-}\leq\alpha^{+}<\min\{p^{-}-1,\frac{q^{\prime}(x)}{p^{\prime}(x)}\}\\ \frac{\gamma}{\theta e}<\beta^{-}\leq\beta^{+}<\min\{q^{-}-1,\frac{p^{\prime}(x)}{q^{\prime}(x)}\}\end{array}\right.$$

$$L^{p(x)}(\Omega ):=\left\{u:\Omega \to \mathbb{R}:u\text{is a measurable and}{\int }_{\Omega }|u(x){|}^{p(x)}dx<+\infty \right\}$$

$$\Vert u{\Vert }_{L^{p(x)}(\Omega )}:=\left\{\lambda >0;{\int }_{\Omega }{\left|\frac{u(x)}{\lambda }\right|}^{p(x)}dx\le 1\right\}.$$

$$W^{1,p(x)}(\Omega ):=\{u\in L^{p(x)}(\Omega );|\nabla u|\in L^{p(x)}(\Omega )\},$$

$$\Vert u{\Vert }_{W^{1,p(x)}(\Omega )}:=\Vert u{\Vert }_{L^{p(x)}(\Omega )}+\Vert \nabla u{\Vert }_{L^{p(x)}(\Omega )}.$$

$$\begin{array}[]{l}W_{0}^{1,p(x)}(\Omega)\hookrightarrow L^{r(x)}(\Omega)\end{array}$$

$$|\log (x)|\le x^{-\alpha }+C{x}^{\theta },$$

$$|log(x+ϵ)|\le x^{\theta }+C$$

$$|\log (x)|\le x^{-\alpha }\text{for}x\in (0,m).$$

$$|\log (x)|\le x^{\theta }\text{for}x\in (M,+\infty ).$$

$$f(x_0)=\frac{\gamma }{\delta }\left(1-\log \left(\frac{\gamma }{\theta \delta }\right)\right),$$

$$\left\{\begin{array}[]{ll}-\Delta_{p(x)}u=H(x,u,v)&\ \mbox{in}\ \Omega,\\ -\Delta_{q(x)}v=G(x,u,v)&\ \mbox{in}\ \Omega,\\ u=v=0&\ \mbox{on}\ \partial\Omega,\end{array}\right.$$

$$|H(x,s,t)|,|G(x,s,t)|\le C,\text{ for all }(x,s,t)\in \Omega \times [0,T]\times [0,S].$$

$$\underline{u}\le \overline{u}\text{and }\underline{v}\le \overline{v}\text{mbox}in\Omega .$$

$$\left\{\begin{array}[]{l}\int_{\Omega}|\nabla\underline{u}|^{p(x)-2}\nabla u\nabla\phi\ dx,\leq\int_{\Omega}H(x,\underline{u},\underline{v})\phi\ dx,\\ \int_{\Omega}|\nabla\underline{v}|^{q(x)-2}\nabla v\nabla\psi\ dx,\leq\int_{\Omega}G(x,\underline{u},\underline{v})\psi\ dx,\end{array}\right.$$

$$\left\{\begin{array}[]{l}\int_{\Omega}|\nabla\overline{u}|^{p(x)-2}\nabla u\nabla\phi\ dx\geq\int_{\Omega}H(x,\overline{u},\overline{v})\phi\ dx,\\ \int_{\Omega}|\nabla\overline{v}|^{p(x)-2}\nabla u\nabla\phi\ dx\geq\int_{\Omega}H(x,\overline{u},\overline{v})\psi\ dx,\end{array}\right.$$

$$\underline{u}(x)\le u(x)\le \overline{u}(x)\text{mbox}and\underline{v}(x)\le v(x)\le \overline{v}(x),\text{ for a.e. }x\in \Omega .$$

$$H_{1}(x,s,t)=\left\{\begin{array}[]{ll}H(x,\underline{u}(x),\underline{v}(x)),&s\leq\underline{u}(x)\\ H(x,s,\underline{v}(x)),&\underline{u}(x)\leq s\leq\overline{u}(x)\ \mbox{and}\ t\leq\underline{v}(x)\\ H(x,s,t),&\underline{u}(x)\leq s\leq\overline{u}(x)\ \mbox{and}\ \underline{v}(x)\leq t\leq\overline{v}(x)\\ H(x,s,\overline{v}(x)),&\underline{u}(x)\leq s\leq\overline{u}(x)\ \mbox{and}\ t\geq\overline{v}(x)\\ H(x,\overline{u}(x),\overline{v}(x)),&s\geq\overline{u}(x)\ \mbox{and}\ t\geq\overline{v}(x).\end{array}\right.$$

$$G_{1}(x,s,t)=\left\{\begin{array}[]{ll}G(x,\underline{u}(x),\underline{v}(x)),&t\leq\underline{v}(x)\\ G(x,s,\underline{v}(x)),&\underline{v}(x)\leq t\leq\overline{v}(x)\ \mbox{and}\ s\leq\underline{u}(x)\\ G(x,s,t),&\underline{v}(x)\leq t\leq\overline{v}(x)\ \mbox{and}\ \underline{u}(x)\leq s\leq\overline{u}(x)\\ G(x,s,\overline{v}(x)),&\underline{v}(x)\leq t\leq\overline{v}(x)\ \mbox{and}\ s\geq\overline{u}(x)\\ G(x,\overline{u}(x),\overline{v}(x)),&t\geq\overline{v}(x).\end{array}\right.$$

$${\gamma }_1(x,s):=-((\underline{u}(x)-s{)}_{+}{)}^l+((s-\overline{u}(x){)}_{+}{)}^l,$$

$${\gamma }_2(x,s):=-((\underline{v}(x)-s{)}_{+}{)}^l+((s-\overline{v}(x){)}_{+}{)}^l.$$

$$\left\{\begin{array}[]{ll}-\Delta_{p(x)}u=H_{2}(x,u,v)&\ \mbox{in}\ \Omega,\\ -\Delta_{q(x)}v=G_{2}(x,u,v)&\ \mbox{in}\ \Omega,\\ u=v=0&\ \mbox{on}\ \partial\Omega,\end{array}\right.$$

Full text

C.O. Alves was partially supported by CNPq/Brazil 304036/2013-7 and INCT-MAT

A. Moussaoui was supported by CNPq/Brazil 402792/2015-7.

1. Introduction and statement of the main results

Let $\Omega\subset{\mathbb{R}}^{N}$ $(N\geq 2)$ be a bounded domain with smooth boundary $\partial\Omega$. We are interested in the following quasilinear system

[TABLE]

which exhibits a singularity at zero through logarithm function. The variable exponents $\alpha(.),\beta(.)$ are positive, the constants $\gamma,\theta>0$ and $\Delta_{p(x)}$ (resp. $\Delta_{q(x)}$) stands for the $p(x)$-Laplacian (resp. $q(x)$-Laplacian) differential operator on $W_{0}^{1,p(x)}(\Omega)$ (resp. $W_{0}^{1,q(x)}(\Omega)$) with $p,q\in C^{1}(\overline{\Omega}),$

[TABLE]

where $p(x)^{\ast}=\frac{Np(x)}{N-p(x)}$ and $q(x)^{\ast}=\frac{Nq(x)}{N-q(x)}$. In the sequel we denote by

[TABLE]

Throughout this paper, we denote by $\mathcal{M}\subset C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ the pair of functions $(u,v)\in C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ such that there is a constant $c>0$, which depends on $u$ and $v$, verifying

[TABLE]

where $d(x):=dist(x,\partial\Omega).$

A weak solution of (1.1) is a pair $(u,v)\in W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega),$ with $u,v$ being positive a.e. in $\Omega$ and satisfying

[TABLE]

for all $(\phi,\psi)\in W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega).$

The study of problems involving variable exponents growth conditions is widely justified with many physical examples and arise from a variety of nonlinear phenomena. They are used in electrorheological fluids as well as in image restorations. For more inquiries on modeling physical phenomena involving $p(x)$-growth condition we refer to [1, 2, 10, 12, 13, 24, 29, 30, 31, 32].

Elliptic problems involving the logarithmic nonlinearity appear in some physical models like in dynamic of thin films of viscous fluids, see for instance [21]. An interesting point regarding these problems comes out from the fact that $-\log x$ is of sign changing and behaving at the origin like the power function $t^{\alpha}$ for $\alpha<0$ with a slow growth. In addition, the logarithmic function is not invariant by scaling which does not occur with the power function. These facts motivated the recent studies in [26], [14] and [21], where the authors considered the scalar semilinear case of (1.1) (that is, $p(x)=q(x)=2$) with constant exponents and by essentially using the linearity of the principal part. We also mention [25], focusing on problem with constant exponents involving nonlinear operator.

The essential point in this work is that the singularity in system (1.1) comes out through logarithmic nonlinearities involving variable exponents growth conditions. According to our knowledge, it is for the first time when such problems are studied. Our main results provide the existence and regularity of (positive) solutions for problem (1.1). They are stated as follows.

Theorem 1**.**

Assume (1.2) holds. Then

**(i): **

If

[TABLE]

problem (1.1) has a solution $(u,v)\in\mathcal{M}$ for all $\theta,\gamma>0$.

**(ii): **

If

[TABLE]

problem (1.1) has a solution $(u,v)\in\mathcal{M}$ for $\gamma$ small enough and for all $\theta>0$.

**(iii): **

If

[TABLE]

problem (1.1) admits a solution $(u,v)\in\mathcal{M}$ for $\gamma$ and $\theta$ small enough.

Theorem 2**.**

Assume (1.2) and that

[TABLE]

holds for all $x\in\Omega$. Then problem (1.1) has a positive solution $(u,v)\in W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$ satisfying (1.3).

The proof of Theorem 1 is done in section 4. Our approach relies on the sub-supersolutions techniques. However, this method in its system version (see [11, p. 269]) does not work for problem (1.1) due to its noncooperative character, which means that the right hand sides of the equations in (1.1) are not necessarily increasing whenever $u$ (resp. $v$) is fixed in the first (resp. second) equation in (1.1). Another reason this approach cannot be directly implemented is the presence of singularities in (1.1). To overcome this difficulties, we disturb problem (1.1) by introducing a parameter $\varepsilon>0$. This gives rise to a regularized system for (1.1), depending on $\varepsilon>0,$ whose study is relevant for our initial problem. We construct a sub-supersolution pair for the regularized system, independent on $\varepsilon,$ and we show the existence of positive family of solutions $(u_{\varepsilon},v_{\varepsilon})\in C^{1,\gamma}(\overline{\Omega})\times C^{1,\gamma}(\overline{\Omega})$, for certain $\gamma\in(0,1)$, through a new result regarding sub-supersolutions for quasilinear competitive (noncooperative) systems involving variable exponents growth conditions (see section 3). Then, a (positive) solution of (1.1) is obtained by passing to the limit as $\varepsilon\rightarrow 0$ essentially relying on the independence on $\varepsilon$ of the upper and lower bounds of the approximate solutions $(u_{\varepsilon},v_{\varepsilon})$ and on Arzelà-Ascoli’s Theorem. An important part of our result lies in the obtaining of the sub and supersolution which cannot be constructed easily. Precisely, this is due to the fact that $p(x)$-Laplacian opeartor is inhomogeneous and in general, it has no first eigenvalue, that is, the infimum of the eigenvalues of $p(x)$-Laplacian equals [math] (see [15]). At this point, the choice of suitable functions with an adjustment of adequate constants is crucial.

The proof of Theorem 2 is done in section 5. It is chiefly based on a Theorem by Rabinowitz (see [28]) which establishes, for each $\varepsilon>0$, the existence of positive solutions $(u_{\varepsilon},v_{\varepsilon})$ for the regularized problem of (1.1) in $W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$. The solution of (1.1) under assumption (1.7) is obtained by passing to the limit as $\varepsilon\rightarrow 0$. This is based on a priori estimates, Hardy-Sobolev Inequality, and Lebesgue’s dominated convergence Theorem.

A significant feature of our existence results concerns the regularity part. In Theorem 1 the regularity of the obtained solution for problem (1.1) is derived through the weak comparison principle and the regularity result in [9].

2. Preliminaries

Let $p\in C(\overline{\Omega})$ with $p(x)>1$ in $\Omega$. Consider the Lebesgue’s space

[TABLE]

which is a Banach space with the Luxemburg norm

[TABLE]

The Banach space $W^{1,p(x)}(\Omega)$ is defined as

[TABLE]

equipped with the norm

[TABLE]

The space $W_{0}^{1,p(x)}(\Omega)$ is defined as closure of $C_{0}^{\infty}(\Omega)$ in $W^{1,p(x)}(\Omega)$ with respect to the norm. The space $W_{0}^{1,p(x)}(\Omega)$ is separable and reflexive Banach spaces when $p^{-}>1$. For a later use, we recall that the embedding

[TABLE]

is compact with $1\leq r(x)<p(x)^{\ast}$.

The next result gives important properties related to the logarithmic nonlinearity.

Lemma 1**.**

1. $(a)$

For each $\alpha,\theta>0$, there is a constant $C$ that depends only on $\alpha$ and $\theta$ such that

[TABLE]

for all $x>0.$ 2. $(b)$

For each $\theta,\epsilon>0$, there is a constant $C$ that depends only on $\epsilon$ and $\theta$ such that

[TABLE]

for all $x\geq 0.$ 3. $(c)$

Let $\gamma,\theta$ and $\delta$ be real numbers. If $\gamma,\theta>0$ and $\delta>\frac{\gamma}{\theta e}$ then the function $f(x)=\gamma x^{\delta}-\theta\log x,x>0$ attains a positive global minimum.

Proof.

With respect to the inequalities we only prove $(a)$ because $(b)$ can be justified similarly. A simple computation provides $\lim_{x\rightarrow 0^{+}}\frac{|\log(x)|}{x^{-\alpha}}=0.$ Thus, there is a small $m>0$ such that

[TABLE]

On the other hand, the limit $\lim_{x\rightarrow+\infty}\frac{|\log(x)|}{x^{\theta}}=0$ implies that there is $M>0$ such that

[TABLE]

Since the function $\frac{|\log(x)|}{x^{\theta}},$ $x>0$ is continuous for all $x>0$, there is a constant, which depends on $\alpha$ and $\theta$, such that $|\log(x)|\leq Cx^{\theta}$ in $[m,M].$ Therefore $|\log(x)|\leq x^{-\alpha}+Cx^{\theta}$ for all $x>0,$ where the constant $C$ depends only on $\alpha$ and $\theta.$

In order to show $(c)$, observe that $f^{{}^{\prime}}(x)=\theta\delta x^{\delta-1}-\gamma/x$. Then, $f$ has a unique critical point at $x_{0}=(\frac{\gamma}{\theta\delta})^{\frac{1}{\delta}}$. Thus, by solving the inequations $f^{{}^{\prime}}(x)>0$ and $f^{{}^{\prime}}(x)<0$ for $x>0,$ it follows that $f$ is increasing on the interval $[x_{0},+\infty)$ and decreasing on $(-\infty,x_{0}].$ By noticing that

[TABLE]

the condition $\delta>\frac{\gamma}{\theta e}$ implies that $f(x_{0})>0,$ which proves the result. ∎

3. Sub-supersolution Theorem

Let us introduce the quasilinear system

[TABLE]

where $H,G:\Omega\times{\mathbb{R}}^{+}\times{\mathbb{R}}^{+}\rightarrow\mathbb{R}$ are Carathéodory functions satisfying the assumption:

**($\mathrm{H,G)}$: **

Given $T,S>0$, there is a constant $C>0$ such that

[TABLE]

The next result is a key point in the proof of Theorem 1.

Theorem 3**.**

Assume that $\mathrm{(H,G)}$ holds and let $\underline{u}\in W_{0}^{1,p(x)}(\Omega)\cap L^{\infty}(\Omega)$ and $\underline{v}\in W_{0}^{1,q(x)}(\Omega)\cap L^{\infty}(\Omega)$ with $\underline{u},\underline{v}\geq 0$ in $\Omega$ and $\overline{u},\overline{v}\in W^{1,\infty}(\Omega)$ such that

[TABLE]

Suppose that

[TABLE]

and

[TABLE]

for all nonnegative functions $(\phi,\psi)\in W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$. Then problem (3.1) has a (positive) solution $(u,v)\in(W_{0}^{1,p(x)}(\Omega)\cap L^{\infty}(\Omega))\times(W_{0}^{1,q(x)}(\Omega)\cap L^{\infty}(\Omega))$ satisfying

[TABLE]

Proof.

The proof is chiefly based on pseudomonotone operator theory. Define the functions

[TABLE]

and

[TABLE]

In what follows, we fix $l\in(0,1)$ with $\min\{p^{-},q^{-}\}>1+l$ and set

[TABLE]

[TABLE]

Using the above functions, let us introduce the auxiliary problem

[TABLE]

where

[TABLE]

and

[TABLE]

By Minty-Browder Theorem (see, e.g., [27]), problem (3.2) has a solution $(u,v)$ in $W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$. Indeed, let $B:E\rightarrow E^{\prime}$ be a function defined by

[TABLE]

where $E$ is the Banach space $W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$ endowed with the norm

[TABLE]

Let us show that the function $B$ satisfies the hypotheses of Minty-Browder Theorem.

i) B is continuous

Let $(u_{n},v_{n})\in E$ be a sequence that converges to $(u,v)$ in $E$. We need to prove that $\|B(u_{n},v_{n})-B(u,v)\|_{E^{{}^{\prime}}}\rightarrow 0.$ To this end, let $(\phi,\psi)\in E$ with $\|(\phi,\psi)\|_{E}\leq 1.$ By Hölder inequality, one has

[TABLE]

Up to a subsequences, we can assume that $\nabla u_{n}(x)\rightarrow\nabla u(x)$ a.e in $\Omega$ and there exists a function $U\in(L^{p(x)}(\Omega))^{N}$ such that $|\nabla u_{n}(x)|\leq U(x)$ a.e in $\Omega.$ Therefore, the Lebesgue’s Dominated Convergence Theorem yields

[TABLE]

Note that

[TABLE]

Then, the continuity and the boundedness of $H$, together with Lebesgue’s Dominated convergence Theorem and Hölder inequality, gives

[TABLE]

On the other hand, we can assume that $u_{n}(x)\rightarrow u(x)$ a.e in $\Omega$ and that exists $w\in L^{p(x)}(\Omega)$ such that $|u_{n}(x)|\leq w(x)$ a.e in $\Omega.$ Arguing as before we get

[TABLE]

and so,

[TABLE]

Hence, the previous reasoning provides

[TABLE]

and

[TABLE]

which justify the continuity of $B.$

ii) B is bounded

Let us show that if $U\subset E$ is a bounded set then $B(U)\subset E^{\prime}$ is bounded. To this end, consider a bounded set $U$ and $(\phi,\psi)\in E$ such that $\|(\phi,\psi)\|\leq 1$. Then, for $(u,v)\in U$ the Hölder inequality gives

[TABLE]

Since $H_{1}(x,u,v)$ is bounded, we derive that

[TABLE]

On the other hand, since

[TABLE]

the Hölder inequality ensures

[TABLE]

From the above arguments we obtain the boundedness of $B.$

iii) B is coercive

Next, we prove that

[TABLE]

Note that

[TABLE]

where $C$ is a positive constant. The triangular inequality and the fact that $(a+b)^{\theta}\leq a^{\theta}+b^{\theta}$ for nonnegative numbers $a$ and $b$ with $\theta\in(0,1)$ give

[TABLE]

Gathering the last inequality with the embeddings

[TABLE]

we derive

[TABLE]

From (3.3), (3.4) and the above inequality we have

[TABLE]

where $C$ is a positive constant. In the same manner, we can see that

[TABLE]

• •

If $\|\nabla u\|_{L^{p(x)}(\Omega)}\geq 1$ and $\|\nabla v\|_{L^{q(x)}(\Omega)}<1$,

[TABLE]

• •

If $\|\nabla u\|_{L^{p(x)}(\Omega)}\geq 1$ and $\|\nabla v\|_{L^{q(x)}(\Omega)}\geq 1$,

[TABLE]

Consider in $E$ a sequence $\{(u_{n},v_{n})\}_{n}$ such that $\|(u_{n},v_{n})\|\rightarrow+\infty.$ Thus $\|\nabla u_{n}\|_{L^{p(x)}(\Omega)}\rightarrow+\infty$ or $\|\nabla v_{n}\|_{L^{q(x)}(\Omega)}\rightarrow+\infty$. Suppose that the first possibility happens and that $\|\nabla u_{n}\|_{L^{p(x)}(\Omega)}\geq 1$ for all $n\in\mathbb{N}.$ Then, we consider two cases:

• •

$\|\nabla u_{n}\|_{L^{p(x)}(\Omega)}\geq 1$ and $\|\nabla v_{n}\|_{L^{q(x)}(\Omega)}<1$ for $n\in\mathbb{N}.$ In this case we have

[TABLE]

• •

$\|\nabla u_{n}\|_{L^{p(x)}(\Omega)}\geq 1$ and $\|\nabla v_{n}\|_{L^{q(x)}(\Omega)}\geq 1$ for $n\in\mathbb{N}.$ In this second case, we have

[TABLE]

Consequently, in both cases studied above, one has

[TABLE]

The other situations regarding $\|\nabla u_{n}\|_{L^{p(x)}(\Omega)}$ and $\|\nabla v_{n}\|_{L^{q(x)}(\Omega)}$ can be handled in much the same way.

iv) B is pseudomonotone

We recall that $B$ is a pseudomonotone operator if $(u_{n},v_{n})\rightharpoonup(u,v)$ in $E$ and

[TABLE]

then

[TABLE]

for all $(\phi,\psi)\in E.$

If $(u_{n},v_{n})\rightharpoonup(u,v)$ then $u_{n}\rightharpoonup u$ and $v_{n}\rightharpoonup v$ in $W^{1,p(x)}(\Omega)$ and $W^{1,q(x)}(\Omega),$ respectively. Since $H_{1}$ and $G_{1}$ are bounded we must have

[TABLE]

and

[TABLE]

Note that

[TABLE]

The previous arguments can be repeated to show that

[TABLE]

[TABLE]

and

[TABLE]

Gathering the above limits together with (3.5), one has

[TABLE]

From the weak convergence, we get

[TABLE]

and

[TABLE]

Therefore

[TABLE]

and

[TABLE]

Using (3.7) and (3.8) in (3.6), the $(S^{+})$ property of the operators $-\Delta_{p(x)}$ and $-\Delta_{q(x)}$ garantees that $u_{n}\rightarrow u$ in $W_{0}^{1,p(x)}(\Omega)$ and $v_{n}\rightarrow v$ in $W_{0}^{1,q(x)}(\Omega)$. Thus, by continuity of $B$, it turn out that

[TABLE]

for all $(\phi,\psi)\in E.$

Finally, from properties I)- IV) we are in a position to apply [27, Theorem 3.3.6] which ensures that $B$ is surjective. Thereby, there exists $(u,v)\in E$ such that

[TABLE]

and in particular, $(u,v)$ is a solution of (3.2).

It remains to prove that

[TABLE]

We only prove the first inequalities in (3.9) because the second ones can be justified similarly. Set $(\phi,\psi):=((u-\overline{u})_{+},0)$. From the definition of $H_{2}$, we obtain

[TABLE]

Therefore

[TABLE]

wich implies that $u\leq\overline{u}$ in $\Omega.$ Using a quite similar argument for $(\phi,\psi):=((\underline{u}-u)_{+},0)$ we get $\underline{u}\leq u$ in $\Omega.$ This completes the proof. ∎

4. Proof of Theorem 1

For every $\varepsilon>0$, let us introduce the auxiliary problem

[TABLE]

Our goal is to show through Theorem 3 that (4.1) has a positive solution $(u_{\epsilon},v_{\epsilon})$. Then, by passing to the limit as $\epsilon\rightarrow 0^{+}$ we get a solution for the original problem (1.1).

Let $\tilde{\Omega}$ be a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial\tilde{\Omega}$ such that $\overline{\Omega}\subset\tilde{\Omega}$ and denote by $\tilde{d}(x)=dist(x,\partial\tilde{\Omega}).$ In [34, Lemma 3.1], the authors have proved that, for $\delta>0$ small enough and for constants $\eta>0$, the function $w\in C^{1}(\overline{\tilde{\Omega}})\cap C_{0}(\tilde{\Omega})$ defined by

[TABLE]

is a subsolution of the problem

[TABLE]

where $\delta>0$ is a number that does not depend on $\eta$ and $\xi=c_{0}\eta^{\frac{1}{p^{+}-1+\tau}}$ with $\tau\in(0,1)$ a fixed number and $c_{0}>0$ is a number depending only on $\delta,\tau,\tilde{\Omega}$ and $p$. Note that

[TABLE]

Given $\lambda>1$, let $\overline{u}$ and $\overline{v}$ in $C^{1}(\overline{\tilde{\Omega}})$ be the unique solutions of problems

[TABLE]

where $\sigma$ is a real constant.

If $\sigma>0$, considering the corresponding function $w$ for $\eta=\lambda^{\sigma}$ and applying the weak maximum principle we get

[TABLE]

where $C_{0},C_{0}^{\prime},C_{1},C_{1}^{\prime}>0$ and $\tau_{1},\tau_{2}\in(0,1)$ are constants that does not depend on $\lambda.$ If $-1<\sigma<0,$ from [18, Lemma 2.1] and for $\lambda$ large, one has

[TABLE]

where $k_{2},k_{2}^{\prime},c_{2}$ and $c_{2}^{\prime}$ are positive constants independent of $\lambda$ . Moreover, by the strong maximum principle there is a constant $c_{0}>0$ (that can depend on $\lambda$) such that

[TABLE]

Now, let $\underline{u}$ and $\underline{v}$ in $C^{1}(\overline{\Omega})$ be the unique solutions of the homogeneous Dirichlet problems

[TABLE]

By [16, Lemma 2.1] and [18], there exist positive constants $k_{0},K_{1}$ and $K_{2}$, independent of $\lambda$, such that

[TABLE]

and

[TABLE]

By the weak maximum principle we have, $\underline{u}\leq\overline{u}$ and $\underline{v}\leq\overline{v}$ in $\overline{\Omega}$ for $\lambda>1$ sufficiently large.

We state the following existence result for the regularized problem (4.1).

Theorem 4**.**

Under assumptions of Theorem 1, there exists $\varepsilon_{0}>0$ such that system (4.1) has a positive solution $(u_{\varepsilon},v_{\varepsilon})\in(W_{0}^{1,p(x)}(\Omega)\cap L^{\infty}(\Omega))\times(W_{0}^{1,q(x)}(\Omega)\cap L^{\infty}(\Omega))$, for all $\varepsilon\in(0,\varepsilon_{0})$. Moreover, it hold

[TABLE]

Proof.

First, let us show that $(\underline{u},\underline{v})$ is a subsolution for problem (4.1) for all $\epsilon\in(0,\epsilon_{0})$. To this end, pick $\varepsilon_{0}\leq 1/2$. Then, from (4.7) and (4.8), for all $\epsilon\in(0,\varepsilon_{0}),$ one has

[TABLE]

and

[TABLE]

for all $\gamma,\theta>0$, provided that $\lambda>1$ is sufficiently large.

Next, we will show that $(\overline{u},\overline{v})$ is a supersolution for problem (4.1) for all $\epsilon\in(0,\epsilon_{0})$. Denote by $\delta^{\prime}:=dist(\partial\tilde{\Omega},\partial\Omega)$ and fix $\varepsilon_{0}\in(0,1).$ By Lemma 1, there are constants $\sigma_{1},\sigma_{2}\in(0,1)$ and $C_{\sigma_{1},\alpha^{+}},C_{\sigma_{2,\beta^{+}}}>0$ such that, for all $\epsilon\in(0,\epsilon_{0})$, one has

[TABLE]

and

[TABLE]

If (1.4) holds, it follows from (4.4), (4.11), (4.12) and for $\sigma>0$ in (4.3), that

[TABLE]

and

[TABLE]

for all $\gamma,\theta>0$, provided that $\lambda$ is large enough.

If (1.5) is satisfied, combining Lemma 1 with (4.5) and (4.6), by (4.11), (4.12) and for $\sigma\in(-1,0)$ in (4.3), we get

[TABLE]

and

[TABLE]

for $\gamma>0$ small enough, for all $\theta>0$ and all $\varepsilon\in(0,\varepsilon_{0})$, provided that $\lambda$ is sufficiently large.

Finally, if (1.6) holds, using (4.4), (4.6), (4.11) and (4.12), for $\sigma\in(-1,0)$ in (4.3), we obtain

[TABLE]

and similarly

[TABLE]

for all $\gamma,\theta>0$ small and all $\varepsilon\in(0,\varepsilon_{0})$, provided that $\lambda>0$ is large enough.

Consequently, it turns out from (4.13), (4.14), (4.15), (4.16), (4.17) and (4.18) that

[TABLE]

and

[TABLE]

for all $\left(\varphi,\psi\right)\in W_{0}^{1,p(x)}\left(\Omega\right)\times W_{0}^{1,q(x)}\left(\Omega\right)$ with $\varphi,\psi\geq 0$. This shows that $(\overline{u},\overline{v})$ is a supersolution for (4.1) for all $\epsilon\in(0,\epsilon_{0})$.

Then, owing to Theorem 3 we conclude that the perturbed problem (4.1) has a solution $(u_{\varepsilon},v_{\varepsilon})\in W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$ within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$, for all $\epsilon\in(0,\varepsilon_{0}).$ Moreover, according to Lemma 1 combined with (4.9) and (4.10), we have that for $\sigma_{1},\sigma_{2}\in(0,1/N)$, there are constants $C_{\sigma_{1}},C_{\sigma_{2}}>0$ such that

[TABLE]

and

[TABLE]

for some positive constants $A_{1}$ and $A_{2}$. Then, thanks to [9, Lemma 2], we deduce that $(u_{\varepsilon},v_{\varepsilon})\in C^{1,\nu}(\overline{\Omega})\times C^{1,\nu}(\overline{\Omega}),$ for certain $\nu\in(0,1)$. ∎

Proof of Theorem 1.

Set $\epsilon:=\frac{1}{n}$ for $n\geq 1/\varepsilon_{0}$. By Theorem 4, we know that there exists a positive solution $(u_{n},v_{n}):=(u_{\frac{1}{n}},v_{\frac{1}{n}})$ bounded in $C^{1,\nu}(\overline{\Omega})\times C^{1,\nu}(\overline{\Omega}),$ for certain $\nu\in(0,1)$, for problem

[TABLE]

Moreover, the property formulated in (4.10) holds true. Employing Arzelà-Ascoli’s theorem, we may pass to the limit in $C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ and the limit functions $(u,v)\in C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ satisfy (1.1) with $(u,v)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$. This completes the proof. ∎

5. Proof of Theorem 2

This section is devoted to the proof of Theorem 2. For $\epsilon>0,$ let consider the regularized problem

[TABLE]

Our demonstration strategy will be to show- by applying the well known result due to Rabinowitz [28]- that, for each $\lambda>0,$ system (5.1) possesses a positive solution $(u_{\epsilon},v_{\epsilon})$ in $W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$, and then derive a solution of (1.1) by taking the limit $\epsilon\rightarrow 0$.

5.1. Existence result for the regularized system

Fix $\epsilon>0$ and for each pair $(f,g)\in L^{p^{\prime}(x)}(\Omega)\times L^{q^{\prime}(x)}(\Omega),$ let consider the auxiliary problem

[TABLE]

Observe that:

• •

$-\log(|g|+\epsilon)\in L^{p^{\prime}(x)}(\Omega)$: Indeed, consider $\theta>0$ such that $0<\theta(p^{\prime})^{+}\leq q^{\prime}(x)$ for all $x\in\overline{\Omega}$. By Lemma 1, one has

[TABLE]

From $(\ref{H1})$ the claim follows.

• •

$(|g(x)|+\epsilon)^{\alpha(x)}\in L^{p^{\prime}(x)}$: By $(\ref{H1})$ notice that

[TABLE]

Since $W^{1,q(x)}(\Omega)\hookrightarrow L^{q^{\prime}(x)}(\Omega)$ the claim is proved.

In the same manner we have $|\log(|f|+\epsilon)|\in L^{q^{\prime}}(\Omega)$ and $(|f|+\epsilon)^{\beta(x)}\in L^{q^{\prime}(x)}$ for all $f\in L^{p^{\prime}(x)}(\Omega)$. Then, on account of the above remarks, the unique solvability of $(u,v)\in W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$ in (5.2) is readily derived from Minty-Browder Theorem. Therefore, the solution operator

[TABLE]

is well defined.

Lemma 2**.**

The operator $\mathcal{T}:\mathbb{R}^{+}\times L^{p^{\prime}(x)}(\Omega)\times L^{q^{\prime}}(\Omega)\rightarrow L^{p^{\prime}(x)}(\Omega)\times L^{q^{\prime}(x)}(\Omega)$ is continuous and compact.

Proof.

Consider a sequence $(\lambda_{n},f_{n},g_{n})\rightarrow(\lambda,f,g)$ in $\mathbb{R}^{+}\times L^{p^{\prime}(x)}(\Omega)\times L^{q^{\prime}(x)}(\Omega)$ and $(u_{n},v_{n}):=\mathcal{T}(\lambda_{n},f_{n},g_{n}).$ Using $u_{n}$ as a test function, one gets

[TABLE]

Since $\{g_{n}\}$ is bounded in $L^{q\prime(x)}(\Omega)$ by Lemma 1, $\{u_{n}\}$ is bounded in $W^{1,p(x)}(\Omega).$ Let $(u,v):=\mathcal{T}(\lambda,f,g).$ Using $u_{n}-u$ as a test function we have

[TABLE]

Note that

[TABLE]

where the constant $C$ does not depend on $n\in\mathbb{N}.$

In the sequel, up to a subsequence, we can assume that $g_{n}(x)\rightarrow g(x)$ a.e in $\Omega$ and $|g_{n}(x)|\leq h$ a.e in $\Omega$ for some $h\in L^{q^{{}^{\prime}}(x)}(\Omega).$ Then, by Lemma 1 and the Lebesgue Theorem, we have

[TABLE]

A similar reasoning leads to

[TABLE]

Since $\{u_{n}\}$ is bounded in $W_{0}^{1,p(x)}(\Omega)$, from (5.4) we deduce that $u_{n}\rightarrow u$ in $W_{0}^{1,p(x)}(\Omega).$ This proves that $\mathcal{T}$ is continuous.

In order to show that $\mathcal{T}$ is compact, it sufficies to prove that $\overline{\mathcal{T}(U)}$ is compact for all $U\subset E$ bounded. At this point, a quite similar argument as above produces the desired conclusion. This completes the proof. ∎

Theorem 5**.**

Under assumptions (1.2) and (1.7), problem (5.1) admits a solution $(u_{\epsilon},v_{\epsilon})$ for all $\epsilon>0$.

Proof.

From Lemma 2 and invoking [28], there is an unbounded continuum $\mathcal{C}$ of solutions of the equation $(u,v)=\mathcal{T}(\lambda,u,v)$, that is, $(\lambda,u,v)\in\mathcal{C}$ is a solution of (5.1).

On the other hand, by Lemma 1 the function $f(x)=\theta x^{\delta}-\gamma\log x,$ for $x>0$, attains a strictly positive minimum if $\delta>\frac{\gamma}{\theta e}$. Since $\alpha^{-},\alpha^{+}>\frac{\gamma}{\theta e},$ it follows that

• •

if $|u|+\epsilon\geq 1,$ then

[TABLE]

where $m_{-}=\min\{-\gamma\log x+\theta x^{\alpha_{-}},x>0\}.$

• •

if $|u|+\epsilon<1,$ then

[TABLE]

where $m_{+}=\min\{-\gamma\log x+\theta x^{\alpha_{+}},x>0\}.$

Therefore, $-\Delta_{p(x)}u\geq m_{1}>0$ where $m_{1}=\min\{m_{-},m_{+}\}$ and, with a quite similar reasoning, we get $-\Delta_{q(x)}v\geq m_{2}>0$ for some $m_{2}>0.$ Thus, by maximum principle, $\mathcal{C}\setminus\{(0,0,0)\}$ must be constituted by strictly positive functions.

Next, we show that the component $\mathcal{C}$ is unbounded with respect to $\lambda\geq 0$. By contradiction, suppose that there is $\lambda^{\star}>0$ such that $(\lambda,u,v)\in\mathcal{C}$ implies that $\lambda\leq\lambda^{\star}.$ Fix $0<\overline{\gamma}\leq\left(\frac{q^{\prime}}{p^{\prime}}\right)^{-}.$ Using $u$ as a test function we get

[TABLE]

and

[TABLE]

where $C$ depends on $\lambda^{\ast},\varepsilon,\gamma$ and $\overline{\gamma}$. Note that

[TABLE]

where $C$ depends on $\theta$ and $\varepsilon.$ Now we will estimate the integral $\int_{\Omega}|u||v|^{\alpha(x)}\ dx$. We have $|v(x)|^{\alpha(x)}\in L^{{p^{{}^{\prime}}(x)}}(\Omega)$. In order to prove this, note that

[TABLE]

The last function belongs to $L^{1}(\Omega)$ because $W^{1,q(x)}(\Omega)\hookrightarrow L^{q^{\star}(x)}(\Omega)\hookrightarrow L^{q^{{}^{\prime}}(x)}(\Omega).$ Thus by Hölder inequality we obtain

[TABLE]

By Hölder inequality and considering all the possibilities for the norms

$\||v|^{\alpha(x)}\|_{L^{p^{{}^{\prime}}}(\Omega)}$ and $\||v|^{\alpha(x)}\|_{L^{\frac{q^{{}^{\prime}}(x)}{\alpha(x)p^{{}^{\prime}}(x)}}(\Omega)}$, we get

[TABLE]

Using the embedding $W^{1,q(x)}(\Omega)\hookrightarrow L^{q^{{}^{\prime}}(x)}(\Omega)$, considering all the possibilities for the norms $\|v\|_{L^{q^{{}^{\prime}}(x)}(\Omega)}$, $\|\nabla u\|_{L^{p(x)}(\Omega)}$, the estimates (5.8), (5.9), (5.10), (5.11) and repeating the arguments for the integral $\int_{\Omega}|u||v|^{\gamma}\ dx$ we obtain

[TABLE]

Thus,

[TABLE]

A similar reasoning leads to

[TABLE]

Since $\alpha^{+}+1<p^{-}$ and $\beta^{+}+1<q^{-}$ it follows that the component $\mathcal{C}$ is bounded, which is absurd. Consequently, $\mathcal{C}$ crosses the set $\{1\}\times L^{p^{{}^{\prime}}(x)}(\Omega)\times L^{q^{{}^{\prime}}(x)}(\Omega)$ and this implies that there is a solution $(u_{\epsilon},v_{\epsilon})$ of (5.1). The proof is completed. ∎

5.2. Passage to the limit

Set $\epsilon=\frac{1}{n}$ in (5.1) with any integer $n\geq 1$. By applying Theorem 5, we know that there exist $u_{\frac{1}{n}}:=u_{n}$ and $v_{\frac{1}{n}}:=v_{n}$ that solve the problem (5.1) with $\epsilon=\frac{1}{n}$.

Claim: The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are bounded in $W_{0}^{1,p(x)}(\Omega)$ and $W_{0}^{1,q(x)}(\Omega),$ respectively and the weak limits (that exist up to a subsequence) are strictly positive in $\Omega.$

First of all, we know that $-\Delta_{p(x)}u_{n}\geq m_{1}>0$ where $m_{1}=\min\{m_{-},m_{+}\}$. If $w_{1}$ denotes the unique positive solution of

[TABLE]

the maximum principle gives

[TABLE]

By the strong maximum principle (see [18, Theorem 1.2]) we have $\frac{w_{1}}{\partial\eta}>0$ where $\eta$ is the inward normal vector of $\partial\Omega.$ Let $\phi_{q^{-}}$ an eigenfunction associated to the first eigenvalue of the operator $(-\Delta_{q^{-}},W_{0}^{1,q^{-}}(\Omega))$. Note that

[TABLE]

where $P_{1}$ is a positive constant that does not depend on $x\in\Omega.$

Denote $\phi_{p^{-}}$ an eigenfunction associated to the first eigenvalue of the operator $(-\Delta_{p^{-}},W_{0}^{1,p^{-}}(\Omega))$. Reasoning as above, we also have $v_{n}\geq w_{2}>0$ and

[TABLE]

where $L_{1}$ is a positive constant that does not depend on $x\in\Omega.$ with $w_{2}$ the unique positive solution of

[TABLE]

Let $\delta\in(0,1)$. By using $u_{n}$ as a test function in its corresponding system of equations and arguing as in the set of inequalities (5.9) and (5.10) we get

[TABLE]

where $C$ is a constant that depends on $\gamma$ and $\theta.$ Hardy-Sobolev inequality (see [22]), together with the embedding $W_{0}^{1,p(x)}(\Omega)\hookrightarrow W_{0}^{1,p^{-}}(\Omega)$ and the relation (5.15), it follows that

[TABLE]

where the constant $C$ does not depend on $n\in\mathbb{N}.$

By (5.16) and using the reasoning that leads to (5.12) and (5.13), we obtain that $(u_{n},v_{n})$ is bounded in $E.$ Passing to a subsequence we have

• •

$u_{n}\rightharpoonup u$ in $W^{1,p(x)}(\Omega),$

• •

$u_{n}\rightarrow u$ in $L^{p(x)}(\Omega),$

• •

$u_{n}\rightarrow u$ a.e in $\Omega,$

• •

$v_{n}\rightharpoonup v$ in $W^{1,p(x)}(\Omega),$

• •

$v_{n}\rightarrow v$ in $L^{p(x)}(\Omega),$

• •

$v_{n}\rightarrow v$ a.e in $\Omega,$

for some pair $(u,v)\in E.$ From the previous pointwise convergence and the relations between $w_{1},$ $u_{n}$ and $w_{2},$ $v_{n},$ we conclude that $u>0$ and $v>0,$ which proves the claim.

Taking $u_{n}$ as a test function and repeating the arguments of the relations (5.3)-(5.7), we get that $u_{n}\rightarrow u$ in $W_{0}^{1,p(x)}(\Omega)$. Notice that the same argument provides that $v_{n}\rightarrow v$ in $W_{0}^{1,q(x)}(\Omega).$

From the previous strong convergence of $u_{n}$ and $v_{n}$, combined with the Lebesgue’s Dominated Convergence Theorem, we obtain

[TABLE]

and

[TABLE]

for all $(\phi,\psi)\in W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$ and the existence of solution is proved.

Acknowledgements

The work was started while the second and the third author were visiting the Federal University of Campina Grande. They thank professor Claudianor Alves and the other members of the department for hospitality.