Existence, positivity and boundedness of solutions for systems of quasilinear elliptic equations

TL;DR

This paper establishes conditions under which solutions to quasilinear elliptic systems involving p-Laplacian and q-Laplacian operators exist, are positive, and are bounded, using fixed point, comparison principles, and iteration methods.

Contribution

It provides new theoretical results on the existence, positivity, and boundedness of solutions for complex quasilinear elliptic systems, combining multiple analytical techniques.

Findings

Existence of solutions under specified conditions

Positivity of solutions established

Solutions are proven to be bounded

Abstract

This article sets forth results on the existence, positivity and boundedness of solutions for quasilinear elliptic systems involving p-Laplacian and q-Laplacian operators. The approach combines Schaefer's fixed point, comparison principle as well as Moser's iteration procedure.

Equations

$$P$$

$$\left\{\begin{array}[]{c}\int_{\Omega}\left|\nabla u\right|^{p-2}\nabla u\nabla\varphi\ dx=\int_{\Omega}f(x,u,v)\varphi\ dx\\ \int_{\Omega}|\nabla v|^{q-2}\nabla v\nabla\psi\ dx=\int_{\Omega}g(x,u,v)\psi\ dx\end{array}\right.$$

$$|sf(x,s,t)|,|tg(x,s,t)|\le k(x)(|s{|}^{\gamma }+|t{|}^{\delta })$$

$$x\mapsto f(x,u(x),v(x))\in L^{p_C^{\prime }}(\Omega ),x\mapsto g(x,u(x),v(x))\in L^{q_C^{\prime }}(\Omega )$$

$$1<C<\min \{\frac{p^{\star }}{p},\frac{q^{\star }}{q}\}.$$

$$\left|sf(x,s,t)\right|\wedge \left|tg(x,s,t)\right|\le k_{p,q}(x)\left(|s{|}^{\alpha +1}|t{|}^{\beta +1}\right)\vee \left(|s{|}^{pC}+|t{|}^{qC}\right),$$

$$\alpha >-1,\beta >-1,\frac{\alpha +1}{p}+\frac{\beta +1}{q}=1.$$

$$|w_1|\wedge |w_2|:=\max \{|w_1|,|w_2|\}\text{ and }|w_1|\vee |w_2|:=\min \{|w_1|,|w_2|\}.$$

$$\left\{\begin{array}[]{l}(f(x,s,t)-f(x,\bar{s},t))(s-\bar{s})\leq 0,\text{ for a.e }x\in\Omega,\\ \text{for all }t\in\mathbb{R}\text{, and all }s,{\overline{s}}\in{\mathbb{R}}\backslash{\mathbb{\{}}0{\mathbb{\}}},\end{array}\right.$$

$$\left\{\begin{array}[]{l}(g(x,s,t)-g(x,s,\bar{t}))(t-\bar{t})\leq 0,\text{ for a.e }x\in\Omega,\\ \text{for all }s\in\mathbb{R}\text{, and all }t,{\overline{t}}\in{\mathbb{R}}\backslash{\mathbb{\{}}0{\mathbb{\}}}.\end{array}\right.\text{ }$$

$$f(x,s,t)\ge a_p(x)s|s{|}^{\hat{\alpha }-1}|t{|}^{\hat{\beta }+1}+b_p(x)s|s{|}^{p-2}$$

$$g(x,s,t)\ge a_q(x)|s{|}^{\hat{\alpha }+1}|t{|}^{\hat{\beta }-1}t+b_q(x)t|t{|}^{q-2},$$

$$\begin{array}[]{l}\hat{\alpha}+1\neq p,\text{ }\hat{\beta}+1\neq q\text{ \ and \ }\dfrac{\hat{\alpha}+1}{p^{\ast}}+\dfrac{\hat{\beta}+1}{q^{\ast}}<1.\end{array}$$

$$\begin{array}[]{l}f(x,s,t)=\dfrac{k_{p,q}(x)}{2}\left(\dfrac{1}{1+|s|^{-\alpha}}+h_{1}(s)\right)|t|^{\beta+1}\end{array}$$

$$\begin{array}[]{l}g(x,s,t)=\dfrac{k_{p,q}(x)}{2}|s|^{\alpha+1}\left(\dfrac{1}{1+|t|^{-\beta}}+h_{2}(t)\right),\end{array}$$

$$h_{1}(s)=\left\{\begin{array}[]{cc}s^{\alpha}&\text{ if }s\geq 1\\ 1&\text{ if }s\leq 1\end{array}\right.,\text{ \ }h_{2}(t)=\left\{\begin{array}[]{cc}t^{\beta}&\text{ if }t\geq 1\\ 1&\text{ if }t\leq 1,\end{array}\right.$$

$$-1<\alpha ,\beta <0,\frac{\alpha +1}{p}+\frac{\beta +1}{q}=1.$$

$${\int }_{\Omega }|u{|}^{\alpha +1}|v{|}^{\beta +1}dx\le \Vert u{\Vert }_p^p+\Vert v{\Vert }_q^q\le \left(\Vert u{\Vert }_p^{pC}+1\right)+\left(\Vert v{\Vert }_q^{qC}+1\right).$$

$$({\int }_{\Omega }uf(x,u,v)dx)\wedge ({\int }_{\Omega }vg(x,u,v)dx)\le const.\left(\Vert u{\Vert }_{1,p}^{pC}+\Vert v{\Vert }_{1,q}^{qC}+1\right).$$

$$B_{\mathrm{p}}^{\sigma ,p}(\Omega )={\left[W^{E(\sigma )+1,p}(\Omega ),W^{E(\sigma ),p}(\Omega )\right]}_{E(\sigma )+1-\sigma ,\mathrm{p}},$$

$$(P_{z,w})\mathcal{\qquad}\left\{\begin{array}[]{ll}-\Delta_{p}z=f(x,u,v)&\hbox{ in }\Omega\\ -\Delta_{q}w=g(x,u,v)&\hbox{ in }\Omega\\ z=w=0&\hbox{ on }\partial\Omega.\end{array}\right.$$

$$(u_n,v_n)\to (u,v)\text{ in }X.$$

$${\mathbf{f}}_n=f(\cdot ,u_n,v_n),{\mathbf{g}}_n=g(\cdot ,u_n,v_n)\mathbf{f}=f(\cdot ,u,v),\mathbf{g}=g(\cdot ,u,v).$$

$${\mathbf{f}}_n\to \mathbf{f}\text{ in }{L}^{p_C^{\prime }}(\Omega )\text{ and }{\mathbf{g}}_n\to \mathbf{g}\text{ in }{L}^{q_C^{\prime }}(\Omega ).$$

$$|{\mathbf{f}}_{n_{k_l}}(x)|\le \underset{\begin{array}{c}-U(x)\le s\le U(x)\\ -V(x)\le t\le V(x)\end{array}}{\sup }\left|f(x,s,t)\right|,\forall l\in \mathbb{N},a.ex\in \Omega .$$

$$\underset{l\to +\infty }{\lim }{\int }_{\Omega }|{\mathbf{f}}_{n_{k_l}}(x)-\mathbf{f}(x){|}^{p_C^{\prime }}dx=0.$$

$$\underset{n\to +\infty }{\lim }{\Vert {\mathbf{f}}_n-\mathbf{f}\Vert }_{p_C^{\prime }}^{p_C^{\prime }}dx=0.$$

$$\Vert z_n-z{\Vert }_{1,p}^p\le {\int }_{\Omega }\left|{\mathbf{f}}_n(x)-\mathbf{f}(x)\right||z_n-z|dx.$$

$$\Vert z_n-z{\Vert }_{1,p}^{p-1}\le c_p{\left({\int }_{\Omega }{\left|{\mathbf{f}}_n(x)-\mathbf{f}(x)\right|}^{p_C^{\prime }}dx\right)}^{1/p_C^{\prime }}.$$

$$|{\mathbf{f}}_{n_k}(x)|,|{\mathbf{g}}_{n_k}(x)|\le const.,\text{ for a.e}.x\in \Omega ,\forall n\in \mathbb{N},.$$

Full text

1. Introduction

Let $\Omega\subset\mathbb{R}^{N}$ $\left(N\geq 2\right)$ be a bounded domain with smooth boundary $\partial\Omega$. Given $1<p,q<N$, we consider the quasilinear elliptic problem

[TABLE]

where $\Delta_{p}$ and $\Delta_{q}$ stand for the $p$-Laplacian and $q$-Laplacian on $W_{0}^{1,p}(\Omega)$ and $W_{0}^{1,q}(\Omega),$ respectively. The nonlinearities $f,g:\Omega\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ in ($P$) are Carathéodory functions, that is, $f(\cdot,s,t),$ $g(\cdot,s,t)$ are measurable for every $(s,t)\in\mathbb{R}^{2}$, and $f(x,\cdot,\cdot),$ $g(x,\cdot,\cdot)$ are continuous for a.e. $x\in\Omega$.

A solution of ($P$) is understood in the weak sense, which means a pair of functions $(u,v)\in W_{0}^{1,p}\left(\Omega\right)\times W_{0}^{1,q}\left(\Omega\right)$ such that

[TABLE]

for all $(\varphi,\psi)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega),$ provided the integrals in the right-hand side of the above equalities exist.

Quasilinear elliptic systems have been quite intensely investigated in the literature with various methods. Among them, in [2, 12, 13, 16, 17], the authors take advantage of the variational structure of the problem to apply variational methods. In [3, 34], some of these methods combined to Nehari manifolds are used. Nonvariational problems also have been widely investigated through topological methods. Namely, we quote Schaefer’s fixed point [24], monotonicity method [10], Leray-Schauder degree theory [8, 9, 40], fixed point index [39], sub-supersolution technics [4, 25, 22] and blow-up method combined with a suitable degree argument [8]. We also mention [20, 23, 7, 28, 19, 37] focusing on the semilinear case of ($P$), that is, when $p=q=2$. It is worth noting that the aforementioned works focus on the following type growth condition

[TABLE]

where $1<\gamma\leq p,$ $1<\delta\leq q.$

In the present paper, we consider the complementary case in which $|sf(x,s,t)|$ and $|tg(x,s,t)|$ satisfy growth condition of type $|s|^{\gamma}+|t|^{\delta},$ $\gamma\in(p,p^{\star}),$ $\delta\in(q,q^{\star})$, where $p^{\star}$ and $q^{\star}$ are the Sobolev critical exponents, that is, $p^{\star}=\frac{Np}{N-p}$ and $q^{\star}=\frac{Nq}{N-q}$. This represents a serious difficulty to overcome, and is rarely handled in the literature. Moreover, the difficulty is even more** **stressed because, on one the hand, no structural assumption is assumed guaranteeing that the Euler functional associated to problem ($P$) is well defined and therefore, the variational method cannot be applied. On the other hand, the sub-supersolution method does not work for problem ($P$) due to of its noncooperative character. This means that generally the functions $f(x,u,\cdot)$ and $g(x,\cdot,v)$ are not necessarily increasing whenever $u,v$ are fixed. It is worth pointing out that no sign condition is required on the right-hand side nonlinearities and so large classes of quasilinear problems involving $p$-Laplacian operator can be incorporated in ($P$).

Throughout this paper, we assume that the nonlinear terms $f$ and $g$ satisfy the following assumptions:

$($**H.1$)$: **

For $(u,v)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega),$

[TABLE]

where $p_{C}^{\prime}=\dfrac{Cp}{Cp-1},$ $q_{C}^{\prime}=\dfrac{Cq}{Cq-1}$ and

[TABLE]

$($**H.2$)$: **

There exists a positive real function $k_{p,q}\in L^{\infty}(\Omega)$ such that

[TABLE]

for a.e. $x\in\Omega$ and all $s,t\in\mathbb{R},$ with

[TABLE]

Here, for any $w_{1},w_{2}\in\mathbb{R}$, we denote

[TABLE]

Our main interest in this work consists in getting solutions of system ($P$) with additional qualitative properties. Namely, we established the existence, positivity and boundedness of nontrivial solutions. Our first main result deals with existence of nontrivial solutions which is stated as follows.

Theorem 1**.**

Under the assumptions $($H.1$)$ and $($H.2$)$ system ($P$) admits at least one nontrivial solution $(u^{\ast},v^{\ast})$ in $C^{1,\sigma}(\overline{\Omega})\times C^{1,\sigma}(\overline{\Omega})$ for certain $\sigma\in(0,1)$.

The proof of Theorem 1 is chiefly based on Schaefer’s fixed point Theorem (see, e.,g. [11, Theorem 4, Section 9.2.2], [33]), which guarantees the existence of a weak solution $(u^{\ast},v^{\ast})$ in $W_{0}^{1,p}\left(\Omega\right)\times W_{0}^{1,q}\left(\Omega\right)$. This required Besov spaces involvement, especially the embeddings from Besov into Sobolev spaces which is one of a significant feature of the present work. Moreover, we prove there exist two constants $\varepsilon_{1}$ and $\Theta$ such that $0<\varepsilon_{1}\leq\|u^{\ast}\|_{1,p}+\|v^{\ast}\|_{1,q}\leq\Theta<+\infty.$ This ensures the nontriviality character of the obtained solution $(u^{\ast},v^{\ast})$ in $W_{0}^{1,p}\left(\Omega\right)\times W_{0}^{1,q}\left(\Omega\right)$.

The $L^{\infty}$-Boundedness for an arbitrary weak solution of problem ($P$) is also provided in the present work. Combined with the regularity result in ** [38]**, it ensures in particular that the obtained solution $(u^{\ast},v^{\ast})$ is bounded in $C^{1,\sigma}(\overline{\Omega})\times C^{1,\sigma}(\overline{\Omega})$ for certain $\sigma\in(0,1)$. Mainly through Moser’s iteration process one can prove the next result.

Theorem 2**.**

Under assumptions $($H.1$)$ and $($H.2$)$, all solutions $(u,v)$ of ($P$) are bounded in $L^{\infty}(\Omega)\times{L^{\infty}(\Omega)}$.

Another main achievement of our work consists to provide a precise sign information on solutions of problem ($P$). In this respect, we establish the existence of a positive solution $(u,v)$ in the sense that both components $u$ and $v$ are positive. Our argument relies on a comparison principle based on fibering method due to Pohozaev. However, additional assumptions on $f$ and $g$ are required and are formulated as follows:

$($**H.3$)$: **

[TABLE]

[TABLE]

$($**H.4$)$: **

There exist functions $a_{p},a_{q}\in L^{\infty}(\Omega),b_{p}\in L^{\delta_{p}}(\Omega)$ and $b_{q}\in L^{\delta_{q}}(\Omega),$ with $\delta_{p}>N/p$, $\delta_{q}>N/q,$ such that

[TABLE]

and

[TABLE]

for a.e. $x\in\Omega,\,$and all $(s,t)\in\mathbb{\mathbb{R}}\times\mathbb{\mathbb{R}}$, with

[TABLE]

The obtained result on positivity property is formulated as follows.

Theorem 3**.**

Assume that $($H.1$)$ - $($H.4$)$ hold. Then problem ($P$) possesses a positive solution $(u,v)$ in $C^{1,\sigma}(\overline{\Omega})\times C^{1,\sigma}(\overline{\Omega})$ for certain $\sigma\in(0,1)$.

We indicate an example showing the applicability of Theorems 1, 2 and 3.

Example 1**.**

Consider the functions $f,g:\Omega\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ defined by

[TABLE]

and

[TABLE]

where $k_{p,q}(\cdot)$ is a bounded positive function in $L^{\infty}(\Omega)$ and

[TABLE]

with

[TABLE]

It is straightforward to check that conditions $($H.1$)$-$($H.4$)$ are verified. Consequently, Theorems 1, 2 and 3 are applicable providing positive and bounded solutions for system ($P$) with equations whose right-hand sides are given through the preceding functions $f$ and $g$.

The rest of the paper is organized as follows. Section 2 contains the existence of nontrivial solutions for problem ($P$). Section 3 deals with the positivity property while section 4 focuses on $L^{\infty}$-boundedness of solutions.

2. Existence of solutions

Given a number $1<p<\infty$, the space $L^{p}(\Omega)$ is endowed with the norm $\left\|u\right\|_{p}=(\int_{\Omega}|u|^{p}\ dx)^{1/p}$, while on $W_{0}^{1,p}\left(\Omega\right)$ we consider the norm $\left\|u\right\|_{1,p}=\left(\int_{\Omega}\left|\nabla u\right|^{p}\ dx\right)^{1/p}$. Throughout this paper, $p^{\prime}=\dfrac{p}{p-1}$ and $p^{\star}=\frac{Np}{N-p}$ are the conjugate and the Sobolev critical exponents, respectively, while $\langle,\,\rangle_{-1,1}$ denotes the duality brackets between the space $W_{0}^{1,p}(\Omega)$ and its topological dual $W^{1,p^{\prime}}(\Omega)$.

Remark 1**.**

Fix $(u,v)$ in $W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega).$ By (1.1), (1.2) together with Young’s and Jensen’s inequalities it holds

[TABLE]

Then, Poincaré’s inequality implies

[TABLE]

Hence, by assumptions (1.1) - (1.2), Sobolev embedding Theorems are applicable.

We will also make use of Besov space $B_{\mathbf{\mathrm{p}}}^{\sigma,p}(\Omega),$ for $1\leq\mathbf{\mathrm{p}}\leq\infty,$ defined as follows

[TABLE]

where $E(\sigma)$ designates the entire part of the real $\sigma$ (see [36]). Note that for a bounded domain $\Omega$ the above definition remains valid for $W_{0}^{s,p}(\Omega)$ instead of $W^{s,p}(\Omega)$.

Lemma 1**.**

The embeddings $B_{\infty}^{1+C,p}(\Omega)\hookrightarrow W_{0}^{1,p}(\Omega)$ and $B_{\infty}^{1+C,q}(\Omega)\hookrightarrow W_{0}^{1,q}(\Omega)$ are compact.

Proof.

Observe that [27, Proposition 4.3] is applicable due to the compactness of the embedding $W_{0}^{E(\sigma)+1,p}(\Omega)\hookrightarrow W_{0}^{E(\sigma),p}(\Omega)$ (see [1, Theorem 6.2] with $\Omega_{0}=\Omega,$ $k=N,$ $j=E(\sigma),$ $\mathbf{\mathrm{p}}=p$ and $m=1$). Thus the embedding $B_{\mathbf{\mathrm{p}}}^{\sigma,p}(\Omega)\hookrightarrow W_{0}^{E(\sigma),p}(\Omega)$ is compact and therefore, the embedding $B_{\infty}^{1,p}(\Omega)$ (resp. $B_{\infty}^{1,q}(\Omega)$) in $W_{0}^{1,p}(\Omega)$ (resp. $W_{0}^{1,q}(\Omega)$) is compact. By the iteration process, we deduce that the embedding $B_{\infty}^{1+C,p}(\Omega)$ (resp. $B_{\infty}^{1+C,q}(\Omega)$) in $W_{0}^{1,p}(\Omega)$ (resp. $W_{0}^{1,q}(\Omega)$) is also compact. ∎

In the sequel, We denote $t^{\pm}:=\max\{0,\pm t\}$ and we set $X:=W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega)$ equipped with the norm $\left\|(u,v)\right\|_{X}=\left\|u\right\|_{1,p}+\left\|v\right\|_{1,q}$.

In this section we focus on the existence of solutions for system ($P$). Our approach is based on the following Schaefer’s fixed point theorem (see e.g., [33, p.29] and [11, chap. 9.2.2]).

Theorem 4**.**

Assume that $T:X\longrightarrow{X}$ is a continuous mapping which is compact on each bounded subset $\mathcal{B}$ of $X.$ Then, either the equation $x=\tau{Tx}$ has a solution for $\tau=1$ or the set of all solution $x$ is unbounded for $0<\tau<1.$

Let $\mathcal{T}:X\rightarrow X$ be the nonlinear operator such that $\mathcal{T}(u,v)=(z,w)$, where $(z,w)$ is required to satisfy

[TABLE]

By $($H.1$)$, the unique solvability of $(z,w)$ in $(\mathcal{P}_{z,w})$ is readily derived from Minty-Browder Theorem (see, e.g, [6]). Thus, the operator $\mathcal{T}$ is well defined.

Lemma 2**.**

Under assumptions $($H.1$)$ and $($H.2$)$ the operator $\mathcal{T}$ is continuous.

Proof.

Let $(u_{n},v_{n})\in X,$ $(z_{n},w_{n})=\mathcal{T}(u_{n},v_{n})$ with

[TABLE]

Set $(z,w)=\mathcal{T}(u,v)$ and

[TABLE]

The continuity of $\mathcal{T}$ follows if we show that

[TABLE]

Let $\{\mathbf{f}_{n_{k}}\}$ be a subsequence. By (2.1), it follows that $u_{n_{k}}\rightarrow u$ in $L^{pC}(\Omega)$ and $v_{n_{k}}\rightarrow v$ in $L^{qC}(\Omega)$. The continuity of $f$ implies $\mathbf{f}_{n_{k}}(x)\rightarrow\mathbf{f}(x)$ for a.e $x$ in $\Omega.$

On the other hand, one can extract subsequences $u_{n_{k_{l}}}$ and $v_{n_{k_{l}}}$ such that $u_{n_{k_{l}}}(x)\rightarrow u(x)$ and $v_{n_{k_{l}}}(x)\rightarrow v(x)$ for a.e $x$ in $\Omega.$ Moreover, there exist positive functions $U\in L^{pC}(\Omega)$ and $V\in L^{qC}(\Omega)$ such that $|u_{n_{k_{l}}}(x)|\leq U(x)$ and $|v_{n_{k_{l}}}(x)|\leq V(x),$ a.e. $x$ in $\Omega$ and all $l\in\mathbb{N}$. Then, the continuity of $f$ gives

[TABLE]

Owing to Lebesgue’s dominated convergence Theorem, we conclude that

[TABLE]

From the Urysohn’s subsequence principle (see, e.g., [29, Proposition A.6, p.179] or [14]), it follows that all the sequence ($\mathbf{f}_{n}$) obeys to

[TABLE]

Multiplying each equation in $(\mathcal{P}_{z,w})$ by $z_{n}-z$ and $w_{n}-w$, respectively, and integrating over $\Omega$, one gets

[TABLE]

By Hölder’s inequality together with the embedding $W_{0}^{1,p}(\Omega)\hookrightarrow L^{pC}(\Omega)$, one can find a constant $c_{p}>0$ such that

[TABLE]

Thanks to Lemma 2.3, we conclude that $z_{n}\rightarrow z$ in $W_{0}^{1,p}(\Omega).$ A quite similar argument provides $w_{n}\rightarrow w$ in $W_{0}^{1,q}(\Omega)$ and therefore, $\mathcal{T}(u_{n},v_{n})\rightarrow\mathcal{T}(u,v)$ in $X.$ This ends the proof. ∎

Lemma 3**.**

Assume $($H.1$)$ and $($H.2$)$ hold. Then $\mathcal{T}$ is compact.

Proof.

For a bounded sequence $(u_{n},v_{n})_{n}$ in $X$ and $\mathbf{f}_{n},\mathbf{g}_{n}$ defined in (2.2), let us show that there exists a subsequence $(u_{n_{k}},v_{n_{k}})_{n_{k}}$ such that

[TABLE]

From (1.1), the embeddings $W_{0}^{1,p}(\Omega)\hookrightarrow L^{pC}(\Omega)$ and $W_{0}^{1,q}(\Omega)\hookrightarrow L^{qC}(\Omega)$ are compact. By Rellich-Kondrachov compactness Theorem, along a relabeled subsequence, $(u_{n},v_{n})_{n}$ converges strongly in $L^{pC}(\Omega)\times L^{qC}(\Omega)$. Consequently, we can extract subsequence $(u_{n_{k}},v_{n_{k}})\rightarrow(\tilde{u},\tilde{v})$ a.e. in $\Omega.$ Exploiting the continuity of $f$ and $g$, we derive that $\mathbf{f}_{n_{k}}$ and $\mathbf{g}_{n_{k}}$ converge to $\tilde{\mathbf{f}}(x)=f(x,\tilde{u},\tilde{v})$ and $\tilde{\mathbf{g}}(x)=g(x,\tilde{u},\tilde{v})$ a.e. in $\Omega$, as well as, $\mathbf{f}_{n_{k}}$, $\mathbf{g}_{n_{k}}$ are bounded in $L^{p_{C}^{\prime}}(\Omega)$ and $L^{q_{C}^{\prime}}(\Omega),$ respectively.

Set $(z_{n_{k}},w_{n_{k}})=\mathcal{T}(u_{n_{k}},v_{n_{k}})$. We claim that $z_{n_{k}}$ is bounded in the Besov space $B_{\infty}^{1+\frac{1}{p-1},p}(\Omega)$ (resp. $B_{\infty}^{1+(p-1),p}(\Omega)$) if $p\geq 2$ (resp. $p<2$). Indeed, we will apply [36] (precisely, (14), (15) in Lemma 1, and (22), (25) in the proof of Theorem 1) to $z_{n_{k}},$ $\mathbf{f}_{n_{k}}$ and $\mathbf{f}_{\infty}={\lim}\mathbf{f}_{n_{k}}$ a.e in $\Omega$.

Let $h$ in $[0,1]$ and $\theta$ in $\mathcal{D}(\Omega)^{N}$. By using [35, (2.8) in Lemma 1.1], there exists a positive constant $c,$ independent of $n_{k}$ and $h,$ such that

[TABLE]

where $Jac(e^{-h\theta})$ denotes the jacobian of the map $\theta\mapsto e^{-\theta}.$ It follows that

[TABLE]

Consequently, for $k$ sufficiently large, $\left\|\mathbf{f}_{n_{k}}-\mathbf{f}_{\infty}\right\|_{p_{C}^{\prime}}$ tends to $0.$ So, there exists a constant $c_{2}>0,$ independent of $h$ and $n,$ such that for $0\leq h\leq 1,$ one has

[TABLE]

Therefore

[TABLE]

and

[TABLE]

which clearly means that $z_{n_{k}}$ is bounded in the Besov space $B_{\infty}^{1+\frac{1}{p-1},p}(\Omega)$ (resp. $B_{\infty}^{1+(p-1),p}(\Omega)$) if $p\geq 2$ (resp. $p<2$). This proves the claim.

Arguing similarly we infer that $w_{n_{k}}$ is bounded in the Besov space $B_{\infty}^{1+\frac{1}{q-1},q}(\Omega)$ (resp. $B_{\infty}^{1+(q-1),q}(\Omega)$) if $q\geq 2$ (resp. $q<2$).

Finally, thanks to Lemma 1, one can extract a subsequence still denoted by $(z_{n},w_{n})$ which converges strongly in $X.$ Thus, the operator $\mathcal{T}$ is compact, ending the proof of Lemma. ∎

Next, to implement Schaefer’s Theorem, let us introduce, for $\tau\in(0,1],$ the auxiliary problem

[TABLE]

According to the definition of the operator $\mathcal{T}$, system $(\mathcal{P}_{\tau})$ may be formulated as $\tau\mathcal{T}{(u,v)}=(u,v).$

Proposition 1**.**

Assume $($H.1$)$ and $($H.2$)$ hold. Given $\tau\in(0,1],$ let $(u_{\tau},v_{\tau})$ be such that $(u_{\tau},v_{\tau})=\tau\mathcal{T}(u_{\tau},v_{\tau}).$ Then there is a constant $\Theta>0,$ independent of $\tau$, such that $\|(u_{\tau},v_{\tau})\|_{X}\leq\Theta$. Moreover, one can find a constant $\varepsilon_{0}>0$ such that system $(\mathcal{P}_{\tau})$ has no solutions on $\partial\mathcal{O}$, where

[TABLE]

Proof.

Arguing by contradiction, let $(u_{\tau},v_{\tau})$ be an unbounded solution of $(\mathcal{P}_{\tau})$ in $X.$ Multiplying the first and the second equation in $(\mathcal{P}_{\tau})$ by $\dfrac{u_{\tau}}{\|u_{\tau}\|_{1,p}^{p}}$ and $\dfrac{v_{\tau}}{\|v_{\tau}\|_{1,q}^{q}}$, respectively, one has

[TABLE]

Employing $($H.2$)$ it follows that

[TABLE]

where $\lambda_{p,q}$ is the first eigenvalue for a nonlinear elliptic system with Dirichlet boundary condition that can be characterized by

[TABLE]

(see [15]). Then, taking $\|u_{\tau}\|_{1,p}^{p}+\|v_{\tau}\|_{1,q}^{q}$ large enough leads to $\tau\dfrac{k_{p,q}}{\lambda_{p,q}}\rightarrow 0,$ which contradicts (2.5). Consequently, there exists a constant $\Theta>0$ such that all solutions $(u_{\tau},v_{\tau})$ of the equation $(u,v)=\tau{T}(u,v),$ with $\tau\in(0,1],$ verify

[TABLE]

Now, we show the second part of the Proposition 1. Set $\Omega_{2}=\Omega\backslash\Omega_{1}$ with

[TABLE]

Then

[TABLE]

where $C_{p}$ and $C_{q}$ are the best constant in the continuous embedding $W_{0}^{1,p}(\Omega)\hookrightarrow L^{pC}(\Omega)$ and $W_{0}^{1,q}(\Omega)\hookrightarrow L^{qC}(\Omega)$. Arguing similarly with the component $v_{\tau},$ one gets

[TABLE]

Then, for any $\tau\in[0,1[$, it follows that

[TABLE]

Setting $\varepsilon_{0}={(2K)^{-1}}{\min\{C_{p}^{p(1-C)},C_{q}^{q(1-C)}\}}$ one derives

[TABLE]

Consequently, according to (2.6) and (2.7), it is readily seen that the solutions set of the equation $(u_{\tau},v_{\tau})=\tau{T}(u_{\tau},v_{\tau})$ verifying $\|(u_{\tau},v_{\tau})\|_{X}=\varepsilon_{0}/2$ or $\|(u_{\tau},v_{\tau})\|_{X}=2\Theta$ is empty. Namely, system $(\mathcal{P}_{\tau})$ doesn’t admit a solution on the boundary $\partial\mathcal{O}$ for all $\tau\in(0,1]$. This ends the proof. ∎

Now we are ready to prove our existence result.

Proof of Theorem 1.

The proof is a consequence of Lemmas 2 and 3 together with Proposition 1. Hence, owing to Theorem 4 one concludes that system ($P$) admits at least a solution $(u^{\star},v^{\star})$ in $X$ satisfying $\varepsilon_{0}\leq\|(u^{\star},v^{\star})\|_{X}\leq\Theta$ for certain positive constants $\varepsilon_{0}$ and $\Theta$. Moreover, regularity results due to Tolksdorf [38] together with Theorem 2 ensure that $(u^{\star},v^{\star})\in C^{1,\sigma}(\overline{\Omega})\times C^{1,\sigma}(\overline{\Omega})$ for certain $\sigma\in(0,1).$ ∎

3. Positivity

In this section, we show the positivity of the obtained solution $(u^{\star},v^{\star})$ stated in Theorem 3. Our approach is chiefly based on comparison arguments. To do so, let us recall the following results due to Pohozaev in [31, Theorems 5.4.2, 5.5.2 and 5.6.1] (see also [32, Theorems 3.4.2, 3.5.2 and 3.6.1]) for the Dirichlet problem

[TABLE]

where $a,b\in L^{\infty}(\Omega)$ and $1<p<q<p^{\star}$.

Proposition 2**.**

Let $\lambda_{1}$ be the first eigenvalue of $-\Delta_{p}$ and let $\phi_{1}$ the corresponding eigenfunction.

$(\mathrm{1})$**: **

Assume $0\leq\lambda<\lambda_{1}$. Then the problem $(\mathcal{P}_{\lambda})$ has at least one positive weak solution $u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega).$ Moreover, there exists $\sigma\in(0,1)$ such that $u\in C_{loc}^{1,\sigma}(\Omega).$

$(\mathrm{2})$**: **

Assume $0\leq\lambda=\lambda_{1}$ and ${\int_{\Omega}}a(x)\phi_{1}^{q}dx<0.$ Then, the problem $(\mathcal{P}_{\lambda})$ has at least one positive weak solution $u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega).$ Moreover, $u\in C_{loc}^{1,\sigma}(\Omega)$ for certain $\sigma\in(0,1).$

$(\mathrm{3})$**: **

Assume $0\leq\lambda_{1}<\lambda$ and ${\int_{\Omega}}a(x)\phi_{1}^{q}dx<0$. Then, there exists $\varepsilon>0$ such that for $\lambda_{1}<\lambda<\lambda_{1}+\varepsilon,$ problem $(\mathcal{P}_{\lambda})$ admits two positive weak solutions in $W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ and each of them belongs to $C_{loc}^{1,\sigma}(\Omega),$ with $\sigma\in(0,1).$

We also recall the following definition of conditional critical point.

Definition 1**.**

([32], [26]) Let $\Sigma$ be a real Banach space and let $h:\Sigma\rightarrow\mathbb{R}$ be a functional such that $h$ is of class $C^{1}(\Sigma\backslash\{0\})$ and $\tilde{h}:\mathbb{R}\backslash\{0\}\times\Sigma\rightarrow\mathbb{R},$ $\tilde{h}(\lambda,v)=h(\lambda{v}).$ Set $S=\{u\in\Sigma:\|u\|=1\}.$

A point $(\lambda,v)\in\mathbb{R}\setminus\{0\}\times{S}$ is a conditional critical point of the function $\tilde{h}$ if

[TABLE]

where $N_{\mathbb{R}\backslash\{0\}\times{S}}(\lambda,v)$ is the normal cone to the set $\mathbb{R}\backslash\{0\}\times{S}$ at the point $(\lambda,v).$

In what follows, we denote by $\lambda_{b_{p}}>0$ the first $p$-Laplacian eigenvalue associated to the weight $b_{p}$

[TABLE]

where functions ${a_{p}}(\cdot)$ and $b_{p}(\cdot)$ are defined in $($H.4$)$. Let $A_{p}$ and $B_{p}$ be the following applications:

[TABLE]

Lemma 4**.**

$A_{p}$* and $B_{p}$ are weakly continuous in $W_{0}^{1,p}(\Omega).$*

Proof.

Let $u$ and $u_{n}$ in $W_{0}^{1,p}(\Omega)$ such that

[TABLE]

We claim that $A_{p}(u_{n})$ tends to $A_{p}(u).$ Indeed, writing

[TABLE]

we distinguish two cases regarding exponents $\hat{\alpha}$ and $\hat{\beta}$.

Case 1: $\dfrac{\hat{\alpha}+1}{p}+\dfrac{\hat{\beta}+1}{q}\geq 1.$

From condition (1.3) there exists a pair $(\theta_{p},\theta_{q})\in]1,p^{\star}[\times]1,q^{\star}[$ such that

[TABLE]

By (3.2), since the embedding $W_{0}^{1,p}(\Omega)\hookrightarrow L^{p\theta_{p}}(\Omega)$ is compact, one gets

[TABLE]

Fatou’s Lemma implies

[TABLE]

Since $a_{p}\in L^{\infty}(\Omega)$, by triangular and Young’s inequalities, we obtain

[TABLE]

Thus, due to (3.2), one derives

[TABLE]

Passing on the upper limit, it follows that

[TABLE]

Hence, (3.3) and (3.4) result in

[TABLE]

Case 2: $0<\dfrac{\hat{\alpha}+1}{p}+\dfrac{\hat{\beta}+1}{q}<1.$

Observe that the argument used in the first case remains valid. Thus

[TABLE]

where ${\hat{r}}=(\dfrac{\hat{\alpha}+1}{p}+\dfrac{\hat{\beta}+1}{q})^{-1}$. Moreover, considering the term

[TABLE]

a quite similar reasoning as above provides

[TABLE]

Thereby, in both cases, we have ${\lim_{n\rightarrow+\infty}}A_{p}(u_{n})=A_{p}(u),$ which proves the claim.

Now, we prove that ${\lim_{n\rightarrow+\infty}}B_{p}(u_{n})=B_{p}(u)$. Write $b_{p}=b_{p}^{+}-b_{p}^{-}$ and proceeding as in the first case, we obtain on the one hand (the result remains the same if we change $b^{+}$ by $b^{-}$)

[TABLE]

and on the other hand

[TABLE]

Hölder’s inequality implies

[TABLE]

Since $\delta_{p}>N/p,$ the embedding $W_{0}^{1,p}(\Omega)\hookrightarrow L^{\delta_{p}^{\prime}p}(\Omega)$ is compact (here $\delta_{p}^{\prime}=\dfrac{\delta_{p}}{\delta_{p}-1}$). Thereby, the sequence $u_{n}$ converges strongly to $u$ in $L^{\delta_{p}^{\prime}p}(\Omega)$ and therefore, the right hand in (3.7) tends to $0.$ Thus, by (3.6), one has

[TABLE]

Combining with (3.5) it follows clearly that

[TABLE]

Similarly, taking $b_{p}^{-}$ instead of $b_{p}^{+},$ one has

[TABLE]

Consequently, the application $B_{p}$ is weakly continuous on $W_{0}^{1,p}(\Omega).$ The proof is achieved. ∎

Lemma 5**.**

Let $\hat{\alpha}\neq{p-1,}$ $a_{p}^{+}\neq 0$ in $\Omega$ and assume $\lambda\geq 0$ such that one of the following conditions is satisfied: either

$($**i$)$: **

$\lambda<\lambda_{b_{p}}$* ; or*

$($**ii$)$: **

$\lambda=\lambda_{b_{p}}$* or $\lambda_{b_{p}}<\lambda<\lambda_{b_{p}}+\varepsilon_{p}$ for a certain $\varepsilon_{p}>0$, and $\int_{\Omega}a_{p}u_{p}^{\hat{\alpha}+1}|v^{\star}|^{\hat{\beta}+1}dx<0,$ where $u_{p}$ is the eigenvalue associated to the first eigenvalue $\lambda_{b_{p}}.$*

Then problem

[TABLE]

admits at least one weak positive solution $\mathcal{U}$ in $W_{0}^{1,p}(\Omega)$.

Proof.

Inspired by [31, sections 3.3 - 3.6], let consider the Euler functional

[TABLE]

where the applications $A_{p}$ and $B_{p}$ are defined in (3.1).

By Definition 1 and under the conditions $t\in\mathbb{R}$ and $\displaystyle\int_{\Omega}|\nabla z|^{p}dx-\lambda{B_{p}(z)}=1,$ one gets

[TABLE]

$E_{p,\lambda}(u)$ becomes

[TABLE]

In addition, the assumption $\hat{\alpha}+1\neq{p}$ ensures from Definition 1 that the conditional critical point of ${\hat{E}}_{p,\lambda}$ is related to the maximization problem

[TABLE]

Thanks to Lemma 4, it is clear that the required assumptions (f0)-(g0) in [31] or (AO)-(BO) in [32] are fulfilled. Consequently, by $($i$)$ or $($ii$)$ in Lemma 5, Proposition 2 ensures that problem (3.8) admits at least one positive weak solution $\mathcal{U}$. ∎

Now, we are ready to prove the positivity result stated in Theorem 3.

Proof of Theorem 3.

Let us define on $\Omega\times W_{0}^{1,p}(\Omega)$ the operator $\mathcal{L}_{p,v^{\star}}$ as follows

[TABLE]

By $($H.4$)$ one has

[TABLE]

In addition, using again $($H.4$)$, we have

[TABLE]

Here, (3.9) and (3.10) should be understood in the weak sense, that is,

[TABLE]

and

[TABLE]

for all $\phi\in W_{0}^{1,p}(\Omega),$ $\phi\geq 0$ in $\Omega.$

We claim that $\mathcal{U}\leq u^{\star}$ in $\Omega.$ Indeed, testing with $(\mathcal{U}-u^{\star})^{+}$ the equation (3.10) and integrating over $\Omega$, one has

[TABLE]

which is equivalent to

[TABLE]

Since

[TABLE]

the monotonicity assumption $($H.3$)$ implies that the right hand side remains negative while the left hand side is positive. Thus $(\mathcal{U}-u^{\star})^{+}=0$ in $\Omega$ forces $\mathcal{U}\leq u^{\star}$ in $\Omega$.

Finally, because $\mathcal{U}>0$ in $\Omega,$ we infer that $u^{\star}>0$ in $\Omega.$ Analogously, we derive that $v^{\star}>0$ in $\Omega$. The proof is complete. ∎

4. Boundedness

Lemma 6**.**

Assume $($H.1$)$ holds. Then, for any solution $(u^{\star},v^{\star})$ of system ($P$), there exists a sequence $(u_{\varepsilon},v_{\varepsilon})\in\left(C^{1}(\bar{\Omega})\cap C^{2}({\Omega})\right)^{2}$ such that

[TABLE]

Proof.

By $($H.1$)$, $f$ and $g$ belong in $L^{p_{C}^{\prime}}(\Omega)$ and $L^{q_{C}^{\prime}}(\Omega),$ respectively. Then, since $C_{0}^{\infty}(\Omega)$ is dense in $L^{p_{C}^{\prime}}(\Omega)$ and $L^{q_{C}^{\prime}}(\Omega),$ one can find a pair $(f_{\varepsilon},g_{\varepsilon})\in C_{0}^{\infty}(\Omega)\times C_{0}^{\infty}(\Omega)$ such that

[TABLE]

Therefore, $f_{\varepsilon}$ admits a subsequence $f_{\varepsilon_{n}}$ which converges a.e in $\Omega.$ Thus, there exists a constant $c_{0}>0$, independent of $\varepsilon_{n},$ such that $\|f_{\varepsilon_{n}}\|_{\infty}\leq c_{0}$ (see, e.g., [30]).

Let $(u_{\varepsilon_{n}},v_{\varepsilon_{n}})$ be a solution defined as follows:

[TABLE]

where $\Delta_{p}^{\varepsilon_{n}}u$ is given by

[TABLE]

It is well known that $(u_{\varepsilon_{n}},v_{\varepsilon_{n}})\in\left(C^{1}(\bar{\Omega})\cap C^{2}({\Omega})\right)^{2}$ (see [21]). Thereby,

[TABLE]

and

[TABLE]

Applying the Hölder’s inequality in the right-hand side of (4.2), the below estimate occurs

[TABLE]

Now, we deal with the left-hand side. For $p>1$, we claim that

[TABLE]

Indeed, by elementary algebra inequality one has

[TABLE]

From (4.3) and (4.4), we deduce

[TABLE]

or again,

[TABLE]

Thus, it’s readily seen that $u_{\varepsilon_{n}}$ is bounded in $W_{0}^{1,p}(\Omega).$ Following the same agrument we obtain that $v_{\varepsilon_{n}}$ is bounded in $W_{0}^{1,q}(\Omega)$. Let $(\tilde{u},\tilde{v})$ be the weak limit of the sequence $(u_{\varepsilon_{n}},v_{\varepsilon_{n}})$ in $W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega)$. The proof is completed by showing that $(\tilde{u},\tilde{v})=(u^{\star},v^{\ast}).$ To this end, let us first show that the strong convergence $u_{\varepsilon_{n}}\rightarrow\tilde{u}$ holds true. Obviously, by weak semi-continuous arguments, it is well known that

[TABLE]

Moreover, consider the application

[TABLE]

and let $\partial\Phi_{\varepsilon}(z)$ be its subdifferential set. Clearly, $\partial\Phi_{\varepsilon}(z)$ is reduced to a single element $-\Delta_{p}^{\varepsilon}(z)$ which is defined on $W_{0}^{1,p}(\Omega)$ as follows

[TABLE]

For all $z\in W_{0}^{1,p}(\Omega),$ we have

[TABLE]

In particular, setting $z=\tilde{u}$ and using (4.4), it follows that

[TABLE]

Passing to the upper-limit on ${\varepsilon}{{}_{n}},$ we get

[TABLE]

Thus, it follows that

[TABLE]

Recalling that $u_{\varepsilon_{n}}\rightharpoonup\tilde{u}$ weakly in $W_{0}^{1,p}(\Omega),$ since $W_{0}^{1,p}(\Omega)$ is an uniform convex Banach space, we conclude that

[TABLE]

Now, we are ready to show that $\tilde{u}=u^{\star}.$ Set $\Phi_{0}(z)={\int_{\Omega}}|\nabla z|^{p}dx$ and denote by $\partial\Phi_{0}(z)$ its subdifferential set. Combining (4.4), (4.5) and passing to the limit on ${\varepsilon}{{}_{n}},$ it follows from (4.6) that

[TABLE]

Since $\partial\Phi_{0}(z)$ contains a single value $-\Delta_{p}z,$ we derive that $f(x,u^{\star},v^{\star})\in\partial\Phi_{0}(\tilde{u})$ and therefore $-\Delta_{p}\tilde{u}=f(x,u^{\star},v^{\star}).$ However, the definition of $u^{\star}$ (see Theorem 1) leads to $-\Delta_{p}\tilde{u}=-\Delta_{p}u^{\star}$ in $\Omega.$ By weak comparison principle, this implies $\tilde{u}=u^{\star}$ in $\Omega$. A quite similar argument produces that $\tilde{v}=u^{\star}$ in $\Omega$, ending the proof. ∎

The next part is devoted to establish the boundedness of the solution $(u^{\star},v^{\star})$.

Lemma 7**.**

For all $k\in\mathbb{N},$ let $(\delta_{k})$ and $(\gamma_{k})$ be the sequences

[TABLE]

where

[TABLE]

Then, the pair $(u_{\varepsilon_{n}},v_{\varepsilon_{n}})$ defined as in (4.1) is bounded in $L^{\delta_{k}}(\Omega)\times L^{\gamma_{k}}(\Omega),$ for all $k\geq 1.$

Proof.

The Lemma is proved if we show that the sequence $(u_{\varepsilon_{n}},v_{\varepsilon_{n}})$ follows the iterative scheme

[TABLE]

Step 1**: **We prove that $(u_{\varepsilon_{n}},v_{\varepsilon_{n}})$ satisfies (4.9) for $k=0$.

Combining (4.7) and (4.8) one has

[TABLE]

Consequently, the embeddings $W_{0}^{1,p}(\Omega)\hookrightarrow L^{\delta_{0}}(\Omega)$ and $W_{0}^{1,q}(\Omega)\hookrightarrow L^{\gamma_{0}}(\Omega)$ are continuous, leading to $u_{\varepsilon_{n}}\in L^{\delta_{0}}(\Omega)$ and $v_{\varepsilon_{n}}\in L^{\gamma_{0}}(\Omega).$

Step 2**: **Let us prove that if $(u_{\varepsilon_{n}},v_{\varepsilon_{n}})\in L^{\delta_{l}}(\Omega)\times L^{\gamma_{l}}(\Omega)$ for $l\in\mathbb{N},$ $l\leq k,$ then $(u_{\varepsilon_{n}},v_{\varepsilon_{n}})\in L^{\delta_{k+1}}(\Omega)\times L^{\gamma_{k+1}}(\Omega).$

For $k\in\mathbb{N},$ we define the sequences $a_{k}$ and $b_{k}$ by

[TABLE]

where constants $C$ and $D$ verify (4.8). Testing the first and the second equations in (4.1) with $u_{\varepsilon_{n}}|u_{\varepsilon_{n}}|^{a_{k}}$ and $v_{\varepsilon_{n}}|v_{\varepsilon_{n}}|^{b_{k}}$, respectively, integrating over $\Omega$, we get

[TABLE]

and

[TABLE]

Clearly, for all $p>1,$ it holds

[TABLE]

Thus, as in (4.4), the left-hand side in (4.10) is estimated by

[TABLE]

Moreover, since the sequence $u_{\varepsilon_{n}}$ belongs to $C^{1}(\overline{\Omega}),$ $u_{\varepsilon_{n}}$ belongs to $W_{0}^{1,p}(\Omega)$ and therefore $u_{\varepsilon_{n}}|u_{\varepsilon_{n}}|^{\frac{a_{k}}{p}}$ belongs to $W_{0}^{1,p}(\Omega).$ Moreover, by $($H.1$)$, one may write $1<pC<p^{\star}$ which ensures that the embedding $W_{0}^{1,p}(\Omega)\hookrightarrow L^{pC}(\Omega)$ is continuous. Hence, there exists a constant $C_{pC}>0$ such that

[TABLE]

However, since

[TABLE]

the iterative inclusion $L^{pC\left(1+\frac{a_{k}}{p}\right)}(\Omega)\subset L^{\delta_{k+1}}(\Omega)$ holds true and then

[TABLE]

or again,

[TABLE]

Gathering (4.12) - (4.15) together**,** the estimate on the left-hand side in (4.10) becomes

[TABLE]

Now, we focus on the right hand side of (4.10). First, we have

[TABLE]

where

[TABLE]

Consequently, there is a constant $R_{k},$ depending on $k,$ such that

[TABLE]

where $t_{k}=DC^{k}+1.$ This means that the inclusion $L^{\delta_{k}}(\Omega)\subset L^{\delta_{k+1}}(\Omega)$ holds true for all $k\geq 1.$ Therefore, since the domain $\Omega$ is bounded, one gets

[TABLE]

Then

[TABLE]

showing that the sequence $u_{\varepsilon_{n}}$ is bounded in every Lebesgue space $L^{\delta_{k}}(\Omega),$ $k\geq 1$. This ends the proof. ∎

Proof of Theorem 3.

From Lemma 6, along a relabelled subsequence still denoted $u_{\varepsilon_{n}}$, we may assume that $u_{\varepsilon_{n}}$ converges a.e. in $\Omega.$ Then, owing to Dominated Convergence Theorem, we infer that

[TABLE]

Again, Dominated Convergence Theorem implies

[TABLE]

By Young’s inequality we get

[TABLE]

Passing to the limit one derives

[TABLE]

By Remark 1, we deduce

[TABLE]

Now, observe that

[TABLE]

Thus, using Young’s inequality on the term $|v^{\star}|^{a_{k}}|v^{\star}|^{qC},$ we get

[TABLE]

Similarly, by considering the component $v_{\varepsilon_{n}},$ we obtain

[TABLE]

Observe that

[TABLE]

and

[TABLE]

Thus

[TABLE]

[TABLE]

that is

[TABLE]

and

[TABLE]

Denote by $E_{k}=\|u^{\star}\|_{\delta_{k}}^{\delta_{k}}+\|v^{\star}\|_{\gamma_{k}}^{\gamma_{k}}.$ Combining (4.19) and (4.20), it follows that

[TABLE]

We set $e_{k}=\ln{E_{k}},$ then we obtain the following iterative scheme

[TABLE]

Proceeding by successive iterations, (4.21) can be formulated as follows

[TABLE]

Then we deduce that

[TABLE]

and

[TABLE]

Else, the estimates hold

[TABLE]

and

[TABLE]

Therefore

[TABLE]

However, recall that $e_{0}=\ln{E_{0}},$ where $E_{0}=\|u^{\star}\|_{\delta_{0}}^{\delta_{0}}+\|v^{\star}\|_{\gamma_{0}}^{\gamma_{0}},$ and so, because the embeddings $W_{0}^{1,p}(\Omega)\hookrightarrow L^{\delta_{0}}(\Omega)$ and $W_{0}^{1,q}(\Omega)\hookrightarrow L^{\gamma_{0}}(\Omega)$ are continuous, more precisely, we also have $E_{0}\leq\|u^{\star}\|_{1,p}^{\delta_{0}}+\|v^{\star}\|_{1,q}^{\gamma_{0}}.$ Since the proof of the first assert in Proposition 1 remains valid by taking $\tau=1,$ then there exists a constant $C_{\Theta},$ depending only on $\Theta,$ such that $E_{0}\leq C_{\Theta}$.

Consequently, the right-hand side in (4.22) is independent of $(u^{\star},v^{\star}).$ The proof is complete. ∎