# Existence of positive solutions for variable exponent elliptic systems

## Abstract

We consider the system of differential equations

where Ω N is a bounded domain with C2 boundary ∂Ω, 1 < p(x) C1 $\left(\stackrel{̄}{\text{Ω}}\right)$ is a function. 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). Many results have been obtained on this kind of problems, for example [1, 38]. In , 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

where Ω N is a bounded domain with C2 boundary Ω, 1 < p(x) C1 $\left(\stackrel{̄}{\text{Ω}}\right)$ is a function. The operator 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

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

is zero in general, and only under some special conditions λ p (x)> 0 (see ), 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 , consider the existence of positive weak solutions for the following p-Laplacian problems

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

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

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

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  investigated the existence of positive solutions of the system

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

where p(x) C1 $\left(\stackrel{̄}{\text{Ω}}\right)$ 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 N is a bounded domain.

To study p(x)-Laplacian problems, we need some theory on the spaces Lp(x)(Ω), W1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see [6, 13]). If Ω N is an open domain, write

Throughout the article, we will assume that:

(H1) Ω N is an open bounded domain with C2 boundary Ω.

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

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

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

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

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

(H5) g : [0, +) (0, +) is a continuous function such that ${L}_{1}={\mathsf{\text{min}}}_{x\in \stackrel{̄}{\text{Ω}}\phantom{\rule{0.3em}{0ex}}}g\left(x\right)$, and ${L}_{2}={\mathsf{\text{max}}}_{x\in \stackrel{̄}{\text{Ω}}}g\left(x\right).$

Denote

We introduce the norm on Lp(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 (Lp(x)(Ω), |.| p (x)) becomes a Banach space, we call it generalized Lebesgue space. The space (Lp(x)(Ω), |.| p (x)) is a separable, reflexive, and uniform convex Banach space (see [, Theorems 1.10 and 1.14]).

The space W1,p(x)(Ω) is defined by W1,p(x)(Ω) = {u Lp(x): | u| Lp(x)}, and it is equipped with the norm

$∥u∥=|u{|}_{p\left(x\right)}+|\nabla u{|}_{p\left(x\right)},\phantom{\rule{1em}{0ex}}\forall u\in {W}^{1,p\left(x\right)}\left(\text{Ω}\right).$

We denote by W01,p(x)(Ω) is the closure of ${C}_{0}^{\infty }\left(\text{Ω}\right)$ in W1,p(x)(Ω). W1,p(x)(Ω) and ${W}_{0}^{1,p\left(x\right)}\left(\text{Ω}\right)$ are separable, reflexive, and uniform convex Banach space (see [, Theorem 2.1] We define

$\left(L\left(u\right),v\right)=\underset{\text{Ω}}{\int }|\nabla u{|}^{p\left(x\right)-2}\nabla u\nabla vdx,\phantom{\rule{1em}{0ex}}\forall v,u\in {W}_{0}^{1,p\left(x\right)}\left(\text{Ω}\right),$

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 [, 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\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\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\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\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

$\begin{array}{c}=\underset{\text{Ω}}{\int }\left(|\nabla u{|}^{p\left(x\right)-2}\nabla u\nabla \phi +l\left(x,\phantom{\rule{2.77695pt}{0ex}}u\right)\phi \right)dx,\\ \forall u\in {W}^{1,p\left(x\right)}\left(\text{Ω}\right),\phantom{\rule{1em}{0ex}}\forall \phi \in {W}_{0}^{1,p\left(x\right)}\left(\text{Ω}\right),\end{array}$

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

Lemma 1.1. (Comparison Principle). Let u, v W1,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),\phantom{\rule{1em}{0ex}}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,\phantom{\rule{1em}{0ex}}\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,\phantom{\rule{1em}{0ex}}2\delta \le d\left(x\right).\hfill \end{array}\right\$

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

(1)

we have the following result

Lemma 1.2. (see ). 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

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

## 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 ), 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,\phantom{\rule{1em}{0ex}}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,\phantom{\rule{1em}{0ex}}\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,\phantom{\rule{1em}{0ex}}2\delta \le d\left(x\right).\hfill \end{array}\right\$

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

By computation

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

(2)

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

$-{\text{Δ}}_{p\left(x\right)}\varphi \le -{k}^{p\left(x\right)}\alpha ,\phantom{\rule{1em}{0ex}}d\left(x\right)<\sigma .$
(3)

Let -λζ = , then

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

from (3), then we have

$-{\text{Δ}}_{p\left(x\right)}\varphi \le {\lambda }^{p\left(x\right)}\left(a\left(0\right){L}_{1}+f\left(0\right)\right)\le {\lambda }^{p\left(x\right)}\left(g\left(x\right)a\left(\varphi \right)+f\left(\varphi \right)\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}d\left(x\right)<\sigma .$
(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

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

then

$-{\text{Δ}}_{p\left(x\right)}\varphi \le {\lambda }^{p\left(x\right)}\left({L}_{1}+1\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\sigma

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),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\sigma
(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),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}2\delta
(6)

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

(7)

Similarly

(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

where $\beta =\beta \left({\lambda }^{{p}^{+}}\left({L}_{2}+1\right)\mu \right)={\mathsf{\text{max}}}_{x\in \stackrel{̄}{\text{Ω}}}{z}_{1}\left(x\right)$. 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 $\gamma {=}_{\delta }^{2}\left({\mathsf{\text{max}}}_{x\in \stackrel{̄}{\text{Ω}}}\varphi \left(x\right)+{\mathsf{\text{max}}}_{x\in \stackrel{̄}{\text{Ω}}}|\nabla \varphi \left(x\right)|\right)$. We claim that

$\varphi \left(x\right)\le {v}_{1}\left(x\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall x\in \text{Ω}.$
(13)

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

and

It only remains to prove that

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

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

Obviously, there exists a positive constant C3 such that

$\gamma \phantom{\rule{0.5em}{0ex}}\le {C}_{\mathsf{\text{3}}}\lambda .$

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

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

${v}_{1}\left(x\right)\le w\left(x\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall x\in \text{Ω}.$
(14)

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

$\varphi \left(x\right)\le {v}_{1}\left(x\right)\le w\left(x\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall x\in \text{Ω}.$
(15)

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

${v}_{1}\left(x\right)\le w\left(x\right)\le {z}_{1}\left(x\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall x\in \text{Ω}.$

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

${\varphi }_{1}\left(x\right)=\varphi \left(x\right)\le {v}_{1}\left(x\right)\le w\left(x\right)\le {z}_{1}\left(x\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall x\in \text{Ω}.$

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

(16)
(17)

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

${C}_{3}\lambda d\left(x\right)\le u\left(x\right)\le {C}_{4}{\left({\lambda }^{{p}^{+}}\left({L}_{2}+1\right)\mu \right)}^{1/\left({p}^{-}-1\right)}{\left(d\left(x\right)\right)}^{\theta },\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}d\left(x\right)\to 0,$
(18)
${C}_{3}\lambda d\left(x\right)\le v\left(x\right)\le {C}_{4}{\left\{{\lambda }^{{p}^{+}}\left({L}_{2}+1\right)h\left[{C}_{2}{\left({\lambda }^{{p}^{+}}\left({L}_{2}+1\right)\mu \right)}^{\frac{1}{{p}^{-}-1}}\right]\right\}}^{\frac{1}{{p}^{-}-1}}{\left(d\left(x\right)\right)}^{\theta },\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}d\left(x\right)\to 0$
(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

It is easy to see

then

Similarly

Thus (16) and (17) are valid.

(ii) Denote

${v}_{3}\left(x\right)=\alpha {\left(d\left(x\right)\right)}^{\theta },\phantom{\rule{1em}{0ex}}d\left(x\right)\le \rho ,$

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

Obviously, v3(x) C1 ρ ), By computation

$-{\text{Δ}}_{p\left(x\right)}{v}_{3}\left(x\right)=-{\left(\alpha \theta \right)}^{p\left(x\right)-1}\left(\theta -1\right)\left(p\left(x\right)-1\right){\left(d\left(x\right)\right)}^{\left(\theta -1\right)\left(p\left(x\right)-1\right)-1}\left(1+\text{Π}\left(x\right)\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}d\left(x\right)<\rho ,$

where

$\text{Π}\left(x\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}d\frac{\left(\nabla p\nabla d\right)\phantom{\rule{0.3em}{0ex}}\mathsf{\text{In}}\phantom{\rule{0.3em}{0ex}}\alpha \theta }{\left(\theta \phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}\left(p\left(x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right)}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}d\frac{\left(\nabla p\nabla d\right)\phantom{\rule{0.3em}{0ex}}\mathsf{\text{In}}\phantom{\rule{0.3em}{0ex}}d}{\left(p\left(x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right)}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}d\frac{\text{Δ}d}{\left(\theta \phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}\left(p\left(x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right)}.$

Let $\alpha \phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\frac{1}{\rho }{C}_{2}{\left({\lambda }^{{p}^{+}}\phantom{\rule{0.3em}{0ex}}\left({L}_{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}\mu \right)}^{1/\left({p}^{-}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right)},$ where ρ > 0 is small enough, it is easy to see that

where ρ > 0 is small enough, then we have

$-\phantom{\rule{0.3em}{0ex}}{\text{Δ}}_{p\left(x\right)}\phantom{\rule{0.3em}{0ex}}{v}_{3}\phantom{\rule{0.3em}{0ex}}\left(x\right)\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}{\lambda }^{{p}^{+}}\phantom{\rule{0.3em}{0ex}}\mu \left({L}_{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right).$

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

$u\left(x\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{C}_{4}\phantom{\rule{0.3em}{0ex}}{\left({\lambda }^{{p}^{+}}\phantom{\rule{0.3em}{0ex}}\left({L}_{2\phantom{\rule{0.3em}{0ex}}}+\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}\mu \right)}^{1/\left({p}^{-}\phantom{\rule{0.3em}{0ex}}-1\right)}\phantom{\rule{0.3em}{0ex}}{\left(d\left(x\right)\right)}^{\theta },\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}as\phantom{\rule{0.3em}{0ex}}d\left(x\right)\phantom{\rule{0.3em}{0ex}}\to \phantom{\rule{0.3em}{0ex}}0.$

Let $\alpha \phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\frac{1}{\rho }{C}_{2}\phantom{\rule{0.3em}{0ex}}{\left\{{\lambda }^{{p}^{+}}\phantom{\rule{0.3em}{0ex}}\left({L}_{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}h\phantom{\rule{0.3em}{0ex}}\left[{C}_{2}\phantom{\rule{0.3em}{0ex}}{\left({\lambda }^{{p}^{+}}\left({L}_{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}\mu \right)}^{\frac{1}{{p}^{-}-1}}\phantom{\rule{0.3em}{0ex}}\right]\phantom{\rule{0.3em}{0ex}}\right\}}^{\frac{1}{{p}^{-}-1}},$ when ρ > 0 is small enough, it is easy to see that

${\left(\alpha \right)}^{p\left(x\right)\phantom{\rule{0.3em}{0ex}}-1}\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}{\lambda }^{{p}^{+}}\phantom{\rule{0.3em}{0ex}}\left({L}_{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}h\phantom{\rule{0.3em}{0ex}}\left[{C}_{2}\phantom{\rule{0.3em}{0ex}}{\left({\lambda }^{{p}^{+}}\phantom{\rule{0.3em}{0ex}}\left({L}_{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}\mu \right)}^{\frac{1}{{p}^{-}-1}}\right]\phantom{\rule{0.3em}{0ex}}.$

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

$v\left(x\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{C}_{4}\phantom{\rule{0.3em}{0ex}}{\left\{{\lambda }^{{p}^{+}\phantom{\rule{0.3em}{0ex}}}\left({L}_{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}h\phantom{\rule{0.3em}{0ex}}\left[{C}_{2}\phantom{\rule{0.3em}{0ex}}{\left({\lambda }^{{p}^{+}}\phantom{\rule{0.3em}{0ex}}\left({L}_{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right)\phantom{\rule{0.3em}{0ex}}\mu \right)}^{\frac{1}{{p}^{-}-1}}\right]\right\}}^{\frac{1}{{p}^{-}-1}}\phantom{\rule{0.3em}{0ex}}{\left(d\left(x\right)\right)}^{\theta }\phantom{\rule{0.3em}{0ex}}as\phantom{\rule{0.3em}{0ex}}d\left(x\right)\phantom{\rule{0.3em}{0ex}}\to \phantom{\rule{0.3em}{0ex}}0$

Obviously, when d(x) < σ, we have

$u\left(x\right),\phantom{\rule{0.3em}{0ex}}v\left(x\right)\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}\varphi \left(x\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{e}^{kd\left(x\right)}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}{C}_{3}\lambda d\left(x\right).$

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

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## Acknowledgements

The authors would like to appreciate the referees for their helpful comments and suggestions. The third author partly supported by the National Science Foundation of China (10701066 & 10971087).

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Ala, S., Afrouzi, G.A., Zhang, Q. et al. Existence of positive solutions for variable exponent elliptic systems. Bound Value Probl 2012, 37 (2012). https://doi.org/10.1186/1687-2770-2012-37 