Existence of positive solutions for variable exponent elliptic systems

Abstract

We consider the system of differential equations

$$\left\{\begin{array}{cc}-{\text{Δ}}_{p\left(x\right)}u={\lambda }^{p\left(x\right)}\left[g\left(x\right)a\left(u\right)+f\left(v\right)\right]\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ -{\text{Δ}}_{p\left(x\right)}v={\lambda }^{p\left(x\right)}\left[g\left(x\right)b\left(v\right)+h\left(u\right)\right]\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ u=v=0\hfill & \hfill \mathsf{\text{on }}\partial \text{Ω},\hfill \end{array}\right.$$

where Ω ⊂ ℝ^N is a bounded domain with C² boundary ∂Ω, 1 < p(x) ∈C¹ $\left(\overline{\text{Ω}}\right)$ is a function. ${\text{Δ}}_{p\left(x\right)}u\mathsf{\text{ = div }}\left(|\nabla u{|}^{p\left(x\right)-2}\nabla u\right)$ is called p(x)-Laplacian. We discuss the existence of positive solution via sub-super solutions without assuming sign conditions on f(0), h(0).

MSC: 35J60; 35B30; 35B40.

1. Introduction

The study of diferential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc., (see[1–3]). Many results have been obtained on this kind of problems, for example [1, 3–8]. In [7], Fan gives the regularity of weak solutions for differential equations with variable exponent. On the existence of solutions for elliptic systems with variable exponent, we refer to [8, 9]. In this article, we mainly consider the existence of positive weak solutions for the system

$$\left(P\right)\left\{\begin{array}{cc}-{\text{Δ}}_{p\left(x\right)}u={\lambda }^{p\left(x\right)}\left[g\left(x\right)a\left(u\right)+f\left(v\right)\right]\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ -{\text{Δ}}_{p\left(x\right)}v={\lambda }^{p\left(x\right)}\left[g\left(x\right)b\left(v\right)+h\left(u\right)\right]\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ u=v=0\hfill & \hfill \mathsf{\text{on }}\partial \text{Ω},\hfill \end{array}\right.$$

where Ω ⊂ ℝ^N is a bounded domain with C² boundary ∂ Ω, 1 < p(x) ∈ C¹ $\left(\overline{\text{Ω}}\right)$ is a function. The operator ${\text{Δ}}_{p\left(x\right)}u\mathsf{\text{ = div }}\left(|\nabla u{|}^{p\left(x\right)-2}\nabla u\right)$ is called p(x)-Laplacian. Especially, if p(x) ≡ p (a constant), (P) is the well-known p-Laplacian system. There are many articles on the existence of solutions for p-Laplacian elliptic systems, for example [5, 10]. Owing to the nonhomogeneity of p(x)-Laplacian problems are more complicated than those of p-Laplacian, many results and methods for p-Laplacian are invalid for p(x)-Laplacian; for example, if Ω is bounded, then the Rayleigh quotient

$${\lambda }_{p\left(x\right)}=\underset{u\in W_0^{1,p\left(x\right)}\left(\text{Ω}\right)\backslash \left\{0\right\}}{\mathsf{\text{inf}}}\frac{{\int }_{\text{Ω}}\frac{1}{p\left(x\right)}|\nabla u{|}^{p\left(x\right)}dx}{{\int }_{\text{Ω}}\frac{1}{p\left(x\right)}|u{|}^{p\left(x\right)}dx}$$

is zero in general, and only under some special conditions λ _{p (x)}> 0 (see [11]), and maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist, but the fact that the first eigenvalue λ _p > 0 and the existence of the first eigenfunction are very important in the study of p-Laplacian problems. There are more difficulties in discussing the existence of solutions of variable exponent problems.

Hai and Shivaji [10], consider the existence of positive weak solutions for the following p-Laplacian problems

$$\left(\mathsf{\text{I}}\right)\left\{\begin{array}{cc}-{\text{Δ}}_{p}u=\lambda f\left(v\right)\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ -{\text{Δ}}_{p}v=\lambda g\left(u\right)\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ u=v=0\hfill & \hfill \mathsf{\text{on }}\partial \text{Ω}\hfill \end{array}\right.$$

the first eigenfunction is used to construct the subsolution of p-Laplacian problems success-fully. On the condition that λ is large enough and

$$\underset{u\to +\infty }{\mathsf{\text{lim}}}\frac{f\left[M{\left(g\left(u\right)\right)}^{\frac{1}{\left(p-1\right)}}\right]}{u^{p-1}}=0,\mathsf{\text{for every}}M>0,$$

the authors give the existence of positive solutions for problem (I).

Chen [5], considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:

$$\left(\mathsf{\text{II}}\right)\left\{\begin{array}{cc}-{\text{Δ}}_{p}u=\lambda f\left(u,v\right)=\lambda u^{\alpha }{v}^{\gamma }\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ -{\text{Δ}}_{q}v=\lambda g\left(u,v\right)=\lambda u^{\delta }{v}^{\beta }\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ u=v=0\hfill & \hfill \mathsf{\text{on }}\partial \text{Ω}\hfill \end{array}\right.$$

the first eigenfunction is used to construct the subsolution of problem(II), the main results are as following

(i) If α, β ≥ 0, γ, δ > 0, θ = (p - 1 - α)(q - 1 - β) - γδ > 0, then problem (II) has a positive weak solution for each λ > 0;

(ii) If θ = 0 and pγ = q(p - 1 - α), then there exists λ₀ > 0 such that for 0 < λ < λ₀, then problem (II) has no nontrivial nonnegative weak solution.

On the p(x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist. Even if the first eigenfunction of p(x)-Laplacian exist, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of p(x)-Laplacian problems. Zhang [12] investigated the existence of positive solutions of the system

$$\left\{\begin{array}{cc}-{\text{Δ}}_{p\left(x\right)}u={\lambda }^{p\left(x\right)}f\left(v\right)\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ -{\text{Δ}}_{p\left(x\right)}v={\lambda }^{p\left(x\right)}g\left(u\right)\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ u=v=0\hfill & \hfill \mathsf{\text{on }}\partial \text{Ω},\hfill \end{array}\right.$$

In this article, we consider the existence of positive solutions of the system

$$\left\{\begin{array}{cc}-{\text{Δ}}_{p\left(x\right)}u={\lambda }^{p\left(x\right)}F\left(x,u,v\right)\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ -{\text{Δ}}_{p\left(x\right)}v={\lambda }^{p\left(x\right)}G\left(x,u,v\right)\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ u=v=0\hfill & \hfill \mathsf{\text{on }}\partial \text{Ω},\hfill \end{array}\right.$$

where p(x) ∈ C¹ $\left(\overline{\text{Ω}}\right)$ is a function, F(x, u, v) = [g(x)a(u) + f(v)], G(x, u, v) = [g(x)b(v) +h(u)], λ is a positive parameter and Ω ⊂ ℝ^N is a bounded domain.

To study p(x)-Laplacian problems, we need some theory on the spaces L^p(x)(Ω), W^1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see [6, 13]). If Ω ⊂ ℝ^N is an open domain, write

$$\begin{array}{c}{C}_{+}\left(\text{Ω}\right)=\left\{h:h\in C\left(\text{Ω}\right),h\left(x\right)>1forx\in \text{Ω}\right\},\\ h^{+}=\underset{x\in \text{Ω}}{\mathsf{\text{sup }}}h\left(x\right),h^{-}=\underset{x\in \text{Ω}}{\mathsf{\text{inf}}}h\left(x\right),\mathsf{\text{for any }}h\in C\left(\text{Ω}\right).\end{array}$$

Throughout the article, we will assume that:

(H₁) Ω ⊂ ℝ^N is an open bounded domain with C² boundary ∂ Ω.

(H₂) p(x) ∈ C¹ $\left(\overline{\text{Ω}}\right)$ and 1 < p^- ≤ p⁺.

(H₃) a, b ∈ C¹([0, ∞)) are nonnegative, nondecreasing functions such that

$$\underset{u\to +\infty }{\mathsf{\text{lim}}}\frac{a\left(u\right)}{u^{p{}^{-}-1}}=0,\underset{u\to +\infty }{\mathsf{\text{lim}}}\frac{b\left(u\right)}{u^{p{}^{-}-1}}=0.$$

(H₄) f, h : [0, +∞) → R are C¹, monotone functions, lim _{u →+∞} f(u) = +∞, lim _{u →+∞} h(u) = +∞, and

$$\underset{u\to +\infty }{\mathsf{\text{lim}}}\frac{f\left[M{\left(h\left(u\right)\right)}^{\frac{1}{\left(p^{-}-1\right)}}\right]}{u^{p^{-}-1}}=0,\forall M>0.$$

(H₅) g : [0, +∞) → (0, +∞) is a continuous function such that $L_1={\mathsf{\text{min}}}_{x\in \overline{\text{Ω}}}g\left(x\right)$, and $L_2={\mathsf{\text{max}}}_{x\in \overline{\text{Ω}}}g\left(x\right).$

Denote

$$L^{p\left(x\right)}\left(\text{Ω}\right)=\left\{u|u\mathsf{\text{is a measurable real - valued function}},\underset{\text{Ω}}{\int }|u\left(x\right){|}^{p\left(x\right)}dx<\infty \right\}.$$

We introduce the norm on L^p(x)(Ω) by

$$|u{|}_{p\left(x\right)}=\mathsf{\text{inf}}\left\{\lambda >0:\underset{\text{Ω}}{\int }{\left|\frac{u\left(x\right)}{\lambda }\right|}^{p\left(x\right)}dx\le 1\right\},$$

and (L^p(x)(Ω), |.| _{p (x)}) becomes a Banach space, we call it generalized Lebesgue space. The space (L^p(x)(Ω), |.| _{p (x)}) is a separable, reflexive, and uniform convex Banach space (see [[6], Theorems 1.10 and 1.14]).

The space W^1,p(x)(Ω) is defined by W^1,p(x)(Ω) = {u ∈ L^p(x): | ∇u| ∈ L^p(x)}, and it is equipped with the norm

$$∥u∥=|u{|}_{p\left(x\right)}+|\nabla u{|}_{p\left(x\right)},\forall u\in W^{1,p\left(x\right)}\left(\text{Ω}\right).$$

We denote by W₀^1,p(x)(Ω) is the closure of $C_0^{\infty }\left(\text{Ω}\right)$ in W^1,p(x)(Ω). W^1,p(x)(Ω) and $W_0^{1,p\left(x\right)}\left(\text{Ω}\right)$ are separable, reflexive, and uniform convex Banach space (see [[6], Theorem 2.1] We define

$$\left(L\left(u\right),v\right)=\underset{\text{Ω}}{\int }|\nabla u{|}^{p\left(x\right)-2}\nabla u\nabla vdx,\forall v,u\in W_0^{1,p\left(x\right)}\left(\text{Ω}\right),$$

then $L:W_0^{1,p\left(x\right)}\left(\text{Ω}\right)\to {\left(W_0^{1,p\left(x\right)}\left(\text{Ω}\right)\right)}^{*}$ is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [[14], Theorem 3.1]).

If $u,v\in W_0^{1,p\left(x\right)}\left(\text{Ω}\right),\left(u,v\right)$ is called a weak solution of (P) if it satisfies

$$\left\{\begin{array}{cc}\hfill \underset{\text{Ω}}{\int }|\nabla u{|}^{p\left(x\right)-2}\nabla u\cdot \nabla qdx=\underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}F\left(x,u,v\right)qdx,\hfill & \hfill \forall q\in W_0^{1,p\left(x\right)}\left(\text{Ω}\right),\hfill \\ \hfill \underset{\text{Ω}}{\int }|\nabla v{|}^{p\left(x\right)-2}\nabla v\cdot \nabla qdx=\underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}G\left(x,u,v\right)qdx,\hfill & \hfill \forall q\in W_0^{1,p\left(x\right)}\left(\text{Ω}\right).\hfill \end{array}\right.$$

Define $A:W^{1,p\left(x\right)}\left(\text{Ω}\right)\to {\left(W_0^{1,p\left(x\right)}\left(\text{Ω}\right)\right)}^{*}$as

$$\begin{array}{c}<Au,\phi >=\underset{\text{Ω}}{\int }\left(|\nabla u{|}^{p\left(x\right)-2}\nabla u\nabla \phi +l\left(x,u\right)\phi \right)dx,\\ \forall u\in W^{1,p\left(x\right)}\left(\text{Ω}\right),\forall \phi \in W_0^{1,p\left(x\right)}\left(\text{Ω}\right),\end{array}$$

where l(x, u) is continuous on $\overline{\text{Ω}}\times ℝ$, and l(x, .) is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [15], we have the following lemma.

Lemma 1.1. (Comparison Principle). Let u, v ∈ W^1,p(x)(Ω) satisfying Au - Av ≥ 0 in ${\left(W_0^{1,p\left(x\right)}\left(\text{Ω}\right)\right)}^{*},\phi \left(x\right)=\mathsf{\text{min}}\left\{u\left(x\right)-v\left(x\right),0\right\}$. If$\phi \left(x\right)\in W_0^{1,p\left(x\right)}\left(\text{Ω}\right)$ (i.e., u ≥ v on ∂ Ω ), then u ≥ v a.e. in Ω.

Here and hereafter, we will use the notation d(x, ∂ Ω) to denote the distance of x ∈ Ω to the boundary of Ω.

Denote d(x) = d(x, ∂ Ω) and $\partial {\text{Ω}}_{ϵ}=\left\{x\in \text{Ω}|d\left(x,\partial \text{Ω}\right)<ϵ\right\}$. Since ∂ Ω is C² regularly, then there exists a constant δ ∈ (0, 1) such that $d\left(x\right)\in C^2\left(\overline{\partial {\text{Ω}}_{3\delta }}\right)$, and |∇d(x)| ≡ 1.

Denote

$$v_1\left(x\right)=\left\{\begin{array}{c}\gamma d\left(x\right),d\left(x\right)<\delta ,\hfill \\ \gamma \delta +\underset{\delta }{\overset{d\left(x\right)}{\int }}\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{p^{-}-1}}{\left(L_1+1\right)}^{\frac{2}{p^{-}-1}}dt,\delta \le d\left(x\right)<2\delta ,\hfill \\ \gamma \delta +\underset{\delta }{\overset{2\delta }{\int }}\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{p^{-}-1}}{\left(L_1+1\right)}^{\frac{2}{p^{-}-1}}dt,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

Obviously, $0\le v_1\left(x\right)\in C^1\left(\overline{\text{Ω}}\right)$. Considering

$$-{\text{Δ}}_{p\left(x\right)}w\left(x\right)=\eta \mathsf{\text{in }}\text{Ω},w=0\mathsf{\text{on }}\partial \text{Ω},$$

(1)

we have the following result

Lemma 1.2. (see [16]). If positive parameter η is large enough and w is the unique solution of (1), then we have

(i) For any θ∈ (0, 1) there exists a positive constant C₁ such that

$$C_1{\eta }^{\frac{1}{p^{+}-1+\theta }}\le \underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}w\left(x\right);$$

(ii) There exists a positive constant C ₂ such that

$$\underset{{}_{x\in \overline{\text{Ω}}}}{\mathsf{\text{max }}}w\left(x\right)\le C_2{\eta }^{\frac{1}{p^{-}-1}}.$$

2. Existence results

In the following, when there be no misunderstanding, we always use C _i to denote positive constants.

Theorem 2.1. On the conditions of (H₁) - (H₅), then (P) has a positive solution when λ is large enough.

Proof. We shall establish Theorem 2.1 by constructing a positive subsolution (Φ₁, Φ₂) and supersolution (z₁, z₂) of (P), such that Φ₁ ≤ z₁ and Φ₂ ≤ z₂. That is (Φ₁, Φ₂) and (z₁, z₂) satisfies

$$\left\{\begin{array}{c}\hfill \underset{\text{Ω}}{\int }|\nabla {\text{Φ}}_1{|}^{p\left(x\right)-2}\nabla {\text{Φ}}_1\cdot \nabla qdx\le \underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}g\left(x\right)a\left({\text{Φ}}_1\right)qdx+\underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}f\left({\text{Φ}}_2\right)qdx,\hfill \\ \hfill \underset{\text{Ω}}{\int }|\nabla {\text{Φ}}_2{|}^{p\left(x\right)-2}\nabla {\text{Φ}}_2\cdot \nabla qdx\le \underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}g\left(x\right)b\left({\text{Φ}}_2\right)qdx+\underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}h\left({\text{Φ}}_1\right)qdx,\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{c}\hfill \underset{\text{Ω}}{\int }|\nabla z_1{|}^{p\left(x\right)-2}\nabla z_1\cdot \nabla qdx\ge \underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}g\left(x\right)a\left(z_1\right)qdx+\underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}f\left(z_2\right)qdx,\hfill \\ \hfill \underset{\text{Ω}}{\int }|\nabla z_2{|}^{p\left(x\right)-2}\nabla z_2\cdot \nabla qdx\ge \underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}g\left(x\right)b\left(z_2\right)qdx+\underset{\text{Ω}}{\int }{\lambda }^{p\left(x\right)}h\left(z_1\right)qdx,\hfill \end{array}\right.$$

for all $q\in W_0^{1,p\left(x\right)}\left(\text{Ω}\right)$ with q ≥ 0. According to the sub-supersolution method for p(x)-Laplacian equations (see [16]), then (P) has a positive solution.

Step 1. We construct a subsolution of (P).

Let σ ∈ (0, δ) is small enough. Denote

$$\varphi \left(x\right)=\left\{\begin{array}{c}{e}^{kd\left(x\right)}-1,d\left(x\right)<\sigma ,\hfill \\ e^{k\sigma }-1+\underset{\sigma }{\overset{d\left(x\right)}{\int }}k{e}^{k\sigma }{\left(\frac{2\delta -t}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}dt,\sigma \le d\left(x\right)<2\delta ,\hfill \\ e^{k\sigma }-1+\underset{\sigma }{\overset{2\delta }{\int }}k{e}^{k\sigma }{\left(\frac{2\delta -t}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}dt,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

It is easy to see that $\varphi \in C^1\left(\overline{\text{Ω}}\right)$. Denote

$$\alpha =\mathsf{\text{min}}\left\{\frac{\mathsf{\text{inf }}p\left(x\right)-1}{4\left(\mathsf{\text{sup}}|\nabla p\left(x\right)|+1\right)},1\right\},\zeta =\mathsf{\text{min}}\left\{a\left(0\right)L_1+f\left(0\right),b\left(0\right)L_1+h\left(0\right),-1\right\}.$$

By computation

$$-{\text{Δ}}_{p\left(x\right)}\varphi =\left\{\begin{array}{c}-k{\left(k{e}^{kd\left(x\right)}\right)}^{p\left(x\right)-1}\left[\left(p\left(x\right)-1\right)+\left(d\left(x\right)+\frac{\mathsf{\text{ln}}k}{k}\right)\nabla p\nabla d+\frac{▵d}{k}\right],d\left(x\right)<\sigma ,\hfill \\ \begin{array}{c}\left\{\frac{1}{2\delta -\sigma }\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}-\left(\frac{2\delta -d}{2\delta -\sigma }\right)\left[\left(\mathsf{\text{ln }}k{e}^{k\sigma }{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}\right)\nabla p\nabla d+\text{Δ}d\right]\right\}\\ \times {\left(k{e}^{k\sigma }\right)}^{p\left(x\right)-1}{{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}}}^{-1}\left(L_1+1\right),\sigma <d\left(x\right)<2\delta ,\hfill \\ 0,2\delta <d\left(x\right).\hfill \end{array}\hfill \end{array}\right.$$

From (H₃) and (H₄), there exists a positive constant M > 1 such that

$$f\left(M-1\right)\ge 1,h\left(M-1\right)\ge 1.$$

Let $\sigma =\frac{1}{k}\mathsf{\text{ln}}M$, then

$$\sigma k=\mathsf{\text{ln }}M.$$

(2)

If k is sufficiently large, from (2), we have

$$-{\text{Δ}}_{p\left(x\right)}\varphi \le -k^{p\left(x\right)}\alpha ,d\left(x\right)<\sigma .$$

(3)

Let -λζ = kα, then

$$k^{p\left(x\right)}\alpha \ge -{\lambda }^{p\left(x\right)}\zeta ,$$

from (3), then we have

$$-{\text{Δ}}_{p\left(x\right)}\varphi \le {\lambda }^{p\left(x\right)}\left(a\left(0\right)L_1+f\left(0\right)\right)\le {\lambda }^{p\left(x\right)}\left(g\left(x\right)a\left(\varphi \right)+f\left(\varphi \right)\right),d\left(x\right)<\sigma .$$

(4)

Since $d\left(x\right)\in C^2\left(\overline{\partial {\text{Ω}}_{3\delta }}\right)$, then there exists a positive constant C₃ such that

$$\begin{array}{c}-{\text{Δ}}_{p\left(x\right)}\varphi \le {\left(k{e}^{k\sigma }\right)}^{p\left(x\right)-1}{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}-1}\hfill \\ .\left|\left\{\frac{2\left(p\left(x\right)-1\right)}{\left(2\delta -\sigma \right)\left(p^{-}-1\right)}-\left(\frac{2\delta -d}{2\delta -\sigma }\right)\left[\left(\mathsf{\text{ln }}k{e}^{k\sigma }{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}\right)\nabla p\nabla d+\text{Δ}d\right]\right\}\right|\\ \le C_3{\left(k{e}^{k\sigma }\right)}^{p\left(x\right)-1}\mathsf{\text{In }}k,\sigma <d\left(x\right)<2\delta .\hfill \end{array}$$

If k is sufficiently large, let -λζ = kα, we have

$$C_3{\left(k{e}^{k\sigma }\right)}^{p\left(x\right)-1}\mathsf{\text{ln }}k=C_3{\left(kM\right)}^{p\left(x\right)-1}\mathsf{\text{ln }}k\le {\lambda }^{p\left(x\right)},$$

then

$$-{\text{Δ}}_{p\left(x\right)}\varphi \le {\lambda }^{p\left(x\right)}\left(L_1+1\right),\sigma <d\left(x\right)<2\delta .$$

Since ϕ (x) ≥ 0 and a, f are monotone, when λ is large enough, then we have

$$-{\text{Δ}}_{p\left(x\right)}\varphi \le {\lambda }^{p\left(x\right)}\left(g\left(x\right)a\left(\varphi \right)+f\left(\varphi \right)\right),\sigma <d\left(x\right)<2\delta .$$

(5)

Obviously

$$-{\text{Δ}}_{p\left(x\right)}\varphi =0\le {\lambda }^{p\left(x\right)}\left(L_1+1\right)\le {\lambda }^{p\left(x\right)}\left(g\left(x\right)a\left(\varphi \right)+f\left(\varphi \right)\right),2\delta <d\left(x\right).$$

(6)

Combining (4), (5), and (6), we can conclude that

$$-{\text{Δ}}_{p\left(x\right)}\varphi \le {\lambda }^{p\left(x\right)}\left(g\left(x\right)a\left(\varphi \right)+f\left(\varphi \right)\right),\mathsf{\text{a}}.\mathsf{\text{e}}.\mathsf{\text{ on }}\text{Ω}.$$

(7)

Similarly

$$-{\text{Δ}}_{p\left(x\right)}\varphi \le {\lambda }^{p\left(x\right)}\left(g\left(x\right)b\left(\varphi \right)+h\left(\varphi \right)\right),\mathsf{\text{a}}.\mathsf{\text{e}}.\mathsf{\text{on }}\text{Ω}.$$

(8)

From (7) and (8), we can see that (ϕ₁, ϕ₂) = (ϕ, ϕ) is a subsolution of (P).

Step 2. We construct a supersolution of (P).

We consider

$$\left\{\begin{array}{cc}-{\text{Δ}}_{p\left(x\right)}{z}_1={\lambda }^{p^{+}}\mu \left(L_2+1\right)\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ \hfill -{\text{Δ}}_{p\left(x\right)}{z}_2={\lambda }^{p^{+}}\left(L_2+1\right)h\left(\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right)\hfill & \hfill \mathsf{\text{in }}\text{Ω},\hfill \\ z_1=z_2=0\hfill & \hfill \mathsf{\text{on }}\partial \text{Ω},\hfill \end{array}\right.$$

where $\beta =\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)={\mathsf{\text{max}}}_{x\in \overline{\text{Ω}}}{z}_1\left(x\right)$. We shall prove that (z₁, z₂) is a supersolution for (p).

For $q\in W_0^{1,p\left(x\right)}\left(\text{Ω}\right)$ with q ≥ 0, it is easy to see that

$$\begin{array}{cc}\hfill \underset{\text{Ω}}{\int }|\nabla z_2{|}^{p\left(x\right)-2}\nabla z_2\cdot \nabla qdx& =\underset{\text{Ω}}{\int }{\lambda }^{p^{+}}\left(L_2+1\right)h\left(\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right)qdx\hfill \\ \ge \underset{\int }{\overset{\text{Ω}}{}}{\lambda }^{p^{+}}{L}_{2}h\left(\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right)qdx+\underset{\text{Ω}}{\int }{\lambda }^{p^{+}}h\left(z_1\right)qdx.\hfill \end{array}$$

(9)

Since ${\mathsf{\text{lim}}}_{u\to +\infty }\frac{f\left[M\left(h\left(u\right)\right)\frac{1}{\left(p^{-}-1\right)}\right]}{u^{p^{-}-1}}=0$,when μ is sufficiently large, combining Lemma 1.2 and (H₃), then we have

$$h\left(\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right)\ge b\left(C_2{\left[{\lambda }^{p^{+}}\left(L_2+1\right)h\left(\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right)\right]}^{\frac{1}{p^{-}-1}}\right)\ge b\left(z_2\right)$$

(10)

Hence

$$\underset{\text{Ω}}{\int }|\nabla z_2{|}^{p\left(x\right)-2}\nabla z_2\cdot \nabla qdx\ge \underset{\text{Ω}}{\int }{\lambda }^{p^{+}}g\left(x\right)b\left(z_2\right)qdx+\underset{\text{Ω}}{\int }{\lambda }^{p^{+}}h\left(z_1\right)qdx.$$

(11)

Also

$$\underset{\text{Ω}}{\int }|\nabla z_1{|}^{p\left(x\right)-2}\nabla z_1\cdot \nabla qdx=\underset{\text{Ω}}{\int }{\lambda }^{p^{+}}\left(L_2+1\right)\mu qdx$$

By (H₃), (H₄), when μ is sufficiently large, combining Lemma 1.2 and (H₃), we have

$$\begin{array}{cc}\hfill \left(L_2+1\right)\mu & \ge \frac{1}{{\lambda }^{p^{+}}}{\left[\frac{1}{C_2}\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right]}^{p^{-}-1}\hfill \\ \ge L_{2}a\left(\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right)+f\left(C_2{\left[{\lambda }^{p^{+}}\left(L_2+1\right)h\left(\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right)\right]}^{\frac{1}{p^{-}-1}}\right).\hfill \end{array}$$

Then

$$\underset{\text{Ω}}{\int }|\nabla z_1{|}^{p\left(x\right)-2}\nabla z_1\cdot \nabla qdx\ge \underset{\text{Ω}}{\int }{\lambda }^{p^{+}}g\left(x\right)a\left(z_1\right)qdx+\underset{\text{Ω}}{\int }{\lambda }^{p^{+}}f\left(z_2\right)qdx.$$

(12)

According to (11) and (12), we can conclude that (z₁, z₂) is a supersolution for (P).

It only remains to prove that ϕ₁ ≤ z₁ and ϕ₂ ≤ z₂.

In the definition of v₁(x), let $\gamma {=}_{\delta }^2\left({\mathsf{\text{max}}}_{x\in \overline{\text{Ω}}}\varphi \left(x\right)+{\mathsf{\text{max}}}_{x\in \overline{\text{Ω}}}|\nabla \varphi \left(x\right)|\right)$. We claim that

$$\varphi \left(x\right)\le v_1\left(x\right),\forall x\in \text{Ω}.$$

(13)

From the definition of v₁, it is easy to see that

$$\varphi \left(x\right)\le 2\underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}\varphi \left(x\right)\le v_1\left(x\right),\mathsf{\text{when }}d\left(x\right)=\delta ,$$

and

$$\varphi \left(x\right)\le 2\underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}\varphi \left(x\right)\le v_1\left(x\right),\mathsf{\text{when }}d\left(x\right)\ge \delta .$$

It only remains to prove that

$$\varphi \left(x\right)\le v_1\left(x\right),\mathsf{\text{when }}d\left(x\right)<\delta .$$

Since $v_1-\varphi \in C^1\left(\overline{\partial {\text{Ω}}_{\delta }}\right),$ then there exists a point $x_0\in \overline{\partial {\text{Ω}}_{\delta }}$such that

$$v_1\left(x_0\right)-\varphi \left(x_0\right)=\underset{{}_{x_0\in \overline{\partial {\text{Ω}}_{\delta }}}}{\mathsf{\text{min}}}\left[v_1\left(x\right)-\varphi \left(x\right)\right].$$

If v₁(x₀) - ϕ(x₀) < 0, it is easy to see that 0 < d(x₀) < δ, and then

$$\nabla v_1\left(x_0\right)-\nabla \varphi \left(x_0\right)=0.$$

From the definition of v₁, we have

$$|\nabla v_1\left(x_0\right)|=\gamma =\frac{2}{\delta }\left(\underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}\varphi \left(x\right)+\underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}|\nabla \varphi \left(x\right)|\right)>|\nabla \varphi \left(x_0\right)|.$$

It is a contradiction to ∇v₁(x₀) - ∇ϕ(x₀) = 0. Thus (13) is valid.

Obviously, there exists a positive constant C₃ such that

$$\gamma \le C_{\mathsf{\text{3}}}\lambda .$$

Since $d\left(x\right)\in C^2\left(\overline{\partial {\text{Ω}}_{3\delta }}\right)$, according to the proof of Lemma 1.2, then there exists a positive constant C₄ such that

$$-{\text{Δ}}_{p\left(x\right)}{v}_1\left(x\right)\le C_{*}{\gamma }^{p\left(x\right)-1+\theta }\le C_4{\lambda }^{p\left(x\right)-1+\theta },\mathsf{\text{a}}.\mathsf{\text{e}}.\mathsf{\text{in }}\text{Ω},\mathsf{\text{where }}\theta \in \left(0,1\right).$$

When $\eta \ge {\lambda }^{p^{+}}$ is large enough, we have

$$-{\text{Δ}}_{p\left(x\right)}{v}_1\left(x\right)\le \eta .$$

According to the comparison principle, we have

$$v_1\left(x\right)\le w\left(x\right),\forall x\in \text{Ω}.$$

(14)

From (13) and (14), when $\eta \ge {\lambda }^{p^{+}}$ and λ ≥ 1 is sufficiently large, we have

$$\varphi \left(x\right)\le v_1\left(x\right)\le w\left(x\right),\forall x\in \text{Ω}.$$

(15)

According to the comparison principle, when μ is large enough, we have

$$v_1\left(x\right)\le w\left(x\right)\le z_1\left(x\right),\forall x\in \text{Ω}.$$

Combining the definition of v₁(x) and (15), it is easy to see that

$${\varphi }_1\left(x\right)=\varphi \left(x\right)\le v_1\left(x\right)\le w\left(x\right)\le z_1\left(x\right),\forall x\in \text{Ω}.$$

When μ ≥ 1 and λ is large enough, from Lemma 1.2, we can see that $\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)$ is large enough, then ${\lambda }^{p^{+}}\left(L_2+1\right)h\left(\beta \left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)\right)$ is large enough. Similarly, we have ϕ₂ ≤ z₂. This completes the proof. □

3. Asymptotic behavior of positive solutions

In this section, when parameter λ → +∞, we will discuss the asymptotic behavior of maximum of solutions about parameter λ, and the asymptotic behavior of solutions near boundary about parameter λ.

Theorem 3.1. On the conditions of (H₁)-(H₅), if (u, v) is a solution of (P) which has been given in Theorem 2.1, then

(i) There exist positive constants C₁ and C₂ such that

$$C_1\lambda \le \underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}u\left(x\right)\le C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}}$$

(16)

$$C_1\lambda \le \underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}v\left(x\right)\le C_2{\left\{{\lambda }^{p^{+}}\left(L_2+1\right)h\left[C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}}\right]\right\}}^{\frac{1}{p^{-}-1}}$$

(17)

(ii) for any θ ∈ (0, 1), there exist positive constants C₃ and C₄ such that

$$C_3\lambda d\left(x\right)\le u\left(x\right)\le C_4{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{1/\left(p^{-}-1\right)}{\left(d\left(x\right)\right)}^{\theta },asd\left(x\right)\to 0,$$

(18)

$$C_3\lambda d\left(x\right)\le v\left(x\right)\le C_4{\left\{{\lambda }^{p^{+}}\left(L_2+1\right)h\left[C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}}\right]\right\}}^{\frac{1}{p^{-}-1}}{\left(d\left(x\right)\right)}^{\theta },asd\left(x\right)\to 0$$

(19)

where μ satisfies (10).

Proof. (i) Obviously, when 2δ ≤ d(x), we have

$$u\left(x\right),v\left(x\right)\ge \varphi \left(x\right)=e^{k\sigma }-1+\underset{\sigma }{\overset{2\delta }{\int }}k{e}^{k\sigma }{\left(\frac{2\delta -t}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}dt\ge -\lambda \frac{\zeta }{\alpha }\underset{\sigma }{\overset{2\delta }{\int }}M{\left(\frac{2\delta -t}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}dt,$$

then there exists a positive constant C₁ such that

$$C_1\lambda \le \underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}u\left(x\right)\mathsf{\text{and}}{C}_1\lambda \le \underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}v\left(x\right).$$

It is easy to see

$$u\left(x\right)\le z_1\left(x\right)\le \underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}{z}_1\left(x\right)\le C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}},$$

then

$$\underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}u\left(x\right)\le C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}}.$$

Similarly

$$\underset{x\in \overline{\text{Ω}}}{\mathsf{\text{max }}}v\left(x\right)\le C_2{\left\{{\lambda }^{p^{+}}\left(L_2+1\right)h\left[C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}}\right]\right\}}^{\frac{1}{p^{-}-1}}$$

Thus (16) and (17) are valid.

(ii) Denote

$$v_3\left(x\right)=\alpha {\left(d\left(x\right)\right)}^{\theta },d\left(x\right)\le \rho ,$$

where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, δ) is small enough.

Obviously, v₃(x) ∈ C¹(Ω _ρ ), By computation

$$-{\text{Δ}}_{p\left(x\right)}{v}_3\left(x\right)=-{\left(\alpha \theta \right)}^{p\left(x\right)-1}\left(\theta -1\right)\left(p\left(x\right)-1\right){\left(d\left(x\right)\right)}^{\left(\theta -1\right)\left(p\left(x\right)-1\right)-1}\left(1+\text{Π}\left(x\right)\right),d\left(x\right)<\rho ,$$

where

$$\text{Π}\left(x\right)=d\frac{\left(\nabla p\nabla d\right)\mathsf{\text{In}}\alpha \theta }{\left(\theta -1\right)\left(p\left(x\right)-1\right)}+d\frac{\left(\nabla p\nabla d\right)\mathsf{\text{In}}d}{\left(p\left(x\right)-1\right)}+d\frac{\text{Δ}d}{\left(\theta -1\right)\left(p\left(x\right)-1\right)}.$$

Let $\alpha =\frac{1}{\rho }{C}_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{1/\left(p^{-}-1\right)},$ where ρ > 0 is small enough, it is easy to see that

$${\left(\alpha \right)}^{p{\left(x\right)}^{-1}}\ge {\lambda }^{p^{+}}\mu \left(L_2+1\right)\mathsf{\text{ and }}\left|\text{Π}\left(x\right)\right|\le \frac{1}{2}.$$

where ρ > 0 is small enough, then we have

$$-{\text{Δ}}_{p\left(x\right)}{v}_3\left(x\right)\ge {\lambda }^{p^{+}}\mu \left(L_2+1\right).$$

Obviously v₃(x) ≥ z₁(x) on ∂ Ω _ρ . According to the comparison principle, we have v₃(x) ≥ z₁ (x) on Ω _ρ. Thus

$$u\left(x\right)\le C_4{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{1/\left(p^{-}-1\right)}{\left(d\left(x\right)\right)}^{\theta },asd\left(x\right)\to 0.$$

Let $\alpha =\frac{1}{\rho }{C}_2{\left\{{\lambda }^{p^{+}}\left(L_2+1\right)h\left[C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}}\right]\right\}}^{\frac{1}{p^{-}-1}},$ when ρ > 0 is small enough, it is easy to see that

$${\left(\alpha \right)}^{p\left(x\right)-1}\ge {\lambda }^{p^{+}}\left(L_2+1\right)h\left[C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}}\right].$$

Similarly, when ρ > 0 is small enough, we have

$$v\left(x\right)\le C_4{\left\{{\lambda }^{p^{+}}\left(L_2+1\right)h\left[C_2{\left({\lambda }^{p^{+}}\left(L_2+1\right)\mu \right)}^{\frac{1}{p^{-}-1}}\right]\right\}}^{\frac{1}{p^{-}-1}}{\left(d\left(x\right)\right)}^{\theta }asd\left(x\right)\to 0$$

Obviously, when d(x) < σ, we have

$$u\left(x\right),v\left(x\right)\ge \varphi \left(x\right)=e^{kd\left(x\right)}-1\ge C_3\lambda d\left(x\right).$$

Thus (18) and (19) are valid. This completes the proof. □