Large Solutions of Quasilinear Elliptic System of Competitive Type: Existence and Asymptotic Behavior

Abstract

We study the existence and asymptotic behavior of positive solutions for a class of quasilinear elliptic systems in a smooth boundary via the upper and lower solutions and the localization method. The main results of the present paper are new and extend some previous results in the literature.

1. Introduction

This paper is concerned with the study of positive boundary blow-up solutions to a quasilinear elliptic system of competitive type:

Δ_𝑝𝑢=𝑎(𝑥)𝑢^𝑎𝑣^𝑏ΔinΩ,_𝑝𝑣=𝑏(𝑥)𝑢^𝑐𝑣^𝑒inΩ,𝑢=𝑣=+∞on𝜕Ω, (1.1) where Ω is a bounded 𝐶² domain of 𝐑^𝐍 and Δ_𝑝 stands for the 𝑝-Laplacian operator defined by Δ_𝑝𝑢=div(|∇𝑢|^𝑝−2∇𝑢),𝑝>1. The exponents 𝑎,𝑏,𝑐,𝑒 verify 𝑎,𝑒>𝑝−1,𝑏,𝑐>0,(𝑎−𝑝+1)(𝑒−𝑝+1)>𝑏𝑐. There exists 𝐶(𝑥),𝐷(𝑥)∈𝐶(Ω,𝑅⁺),𝛾(𝑥),𝜂(𝑥)∈𝐶(Ω,𝑅⁺) such that lim_𝑥→𝑥0𝑎(𝑥)𝐶(𝑥₀)𝑑(𝑥)^𝛾(𝑥0)=1,lim_𝑥→𝑥0𝑏(𝑥)𝐷(𝑥₀)𝑑(𝑥)^𝜂(𝑥0)=1, (1.2) where 𝑥₀∈𝜕Ω,𝑑(𝑥)=dist(𝑥,𝜕Ω).

We must emphasize that the weight functions 𝑎(𝑥),𝑏(𝑥) are allowed decaying to zero on Ω with arbitrary rate, depending upon the particular point of 𝜕Ω. The boundary condition is to be understood 𝑢(𝑥)→∞,𝑣(𝑥)→∞ as 𝑑(𝑥)→0⁺. Problems like (1.1) are usually known in the literature as boundary blow-up problems, and their solutions are also named large solutions or boundary blow-up solutions.

The problem of the previous form is mathematical models occuring in studies of the 𝑝-Laplace system, generalized reaction-diffusion theory, non-Newtonian fluid theory [1, 2], non-Newtonian filtration [3], and the turbulent flow of a gas in porous medium. In the non-Newtonian fluid theory, the quantity 𝑝 is a characteristic of the medium. Media with 𝑝>2 are called dilatant fluids and those with 𝑝<2 are called pseudoplastics. If 𝑝=2, they are Newtonian fluids. When 𝑝≠2, the problem becomes more complicated since certain nice properties inherent to the case 𝑝=2 seem to be lost or at least difficult to verify. The main differences between 𝑝=2 and 𝑝≠2 can be founded in [4, 5].

When 𝑝=2, system (1.1) becomes Δ𝑢=𝑎(𝑥)𝑢^𝑎𝑣^𝑏inΩ,Δ𝑣=𝑏(𝑥)𝑢^𝑐𝑣^𝑒inΩ,𝑢=𝑣=+∞on𝜕Ω, (1.3) for which the existence, uniqueness, and asymptotic behavior of large solutions have been investigated extensively. We list here, for example, [6–12].

This is a huge amount of literature dealing with single equation with infinite boundary conditions (see, e.g., [13–34]). This problem with more general nonlinearies and weight-function has been discussed by many authors recently [35–39].

Problem (1.1) is considered in special case. When 𝑝=2, in [40], problem (1.1) was analyzed with 𝑎(𝑥)=1,𝑏(𝑥)=1. In the same paper, some existence, uniqueness, and boundary behavior of solutions were obtained under the assumptions

𝑎(𝑥)∼𝐶₁𝑑(𝑥)^𝑘1,𝑏(𝑥)∼𝐶₂𝑑(𝑥)^𝑘2 (1.4) as 𝑑(𝑥)→0⁺ for some positive constants 𝐶₁,𝐶₂ and real numbers 𝑘₁,𝑘₂>−2. This problem was later studied in [41] with general form, where

𝐶₁𝑑(𝑥)^𝛾1≤𝑎(𝑥)≤𝐶₂𝑑(𝑥)^𝛾1,𝐶^′₁𝑑(𝑥)^𝛾1≤𝑏(𝑥)≤𝐶^′₂𝑑(𝑥)^𝛾1, (1.5) for 𝑥∈Ω, 𝛾₁,𝛾₂∈𝐑^𝐍,𝐶₁,𝐶₂,𝐶^′₁,𝐶^′₂ are positive constants. The author also obtained uniqueness results.

In [42], Yang extended the quasilinear elliptic system to

Δ_𝑝𝑢=𝑢^𝑚1𝑣^𝑛1ΔinΩ,_𝑞𝑣=𝑢^𝑚2𝑣^𝑛2inΩ,𝑢=𝑣=+∞on𝜕Ω, (1.6) where 𝑚₁>𝑝−1,𝑛₂>𝑞−1,𝑚₂,𝑛₁>0, and Ω⊆𝐑^𝐍 is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions:𝑢=𝜆,𝑣=𝜇 or 𝑢=𝑣=+∞ or 𝑢=+∞,𝑣=𝜇 on 𝜕Ω, where 𝜆,𝜇>0. Under several hypotheses on the parameters 𝑚₁,𝑛₁,𝑚₂,𝑛₂, the author showed the existence of positive solutions and further provided the asymptotic behavior of the solutions near 𝜕Ω.

When 𝑝≠2, in [43], problem (1.1) was analyzed with 𝑎(𝑥)=1,𝑏(𝑥)=1 under assumption (1.4). The author obtained the existence, uniqueness, and behavior of solutions to problem (1.1).

Very recently, Huang et al. [12] obtained existence, uniqueness, and asymptotic behavior of problem (1.1) when 𝑝=2, and 𝑎(𝑥),𝑏(𝑥) satisfy condition (1.2). Motivated by the results of the papers [12, 40, 41, 43], we consider the quasilinear elliptic system (1.1). We modify the method developed by Huang et al. [12] and extend the results to a quasilinear elliptic system (1.1) under condition (1.2).

Throughout of this paper, set 𝐶₁=min_𝑥∈Ω𝐶(𝑥),𝐶₂=max_𝑥∈Ω𝐶(𝑥),𝐷₁=min_𝑥∈Ω𝐷(𝑥),𝐷₂=max_𝑥∈Ω𝛾𝐷(𝑥),₁=max_𝑥∈Ω𝛾(𝑥),𝛾₂=min_𝑥∈Ω𝛾(𝑥),𝜂₁=max_𝑥∈Ω𝜂(𝑥),𝜂₂=min_𝑥∈Ω𝜂(𝑥),𝛼(𝑥,𝑦)=(𝑝+𝑥)(𝑒−𝑝+1)−(𝑝+𝑦)𝑏(𝑎−𝑝+1)(𝑒−𝑝+1)−𝑏𝑐,𝛽(𝑥,𝑦)=(𝑝+𝑦)(𝑎−𝑝+1)−(𝑝+𝑥)𝑐,(𝑎−𝑝+1)(𝑒−𝑝+1)−𝑏𝑐𝐸(𝑥,𝑦)=(((𝑝−1)𝛼^𝑝−1(𝛼+1))^{𝑒−𝑝+1}𝑥^𝑏((𝑝−1)𝛽^𝑝−1(𝛽+1))^𝑏𝑦^{𝑒−𝑝+1})^{1/((𝑎−𝑝+1)(𝑒−𝑝+1)−𝑏𝑐)},𝐹(𝑥,𝑦)=(((𝑝−1)𝛽^𝑝−1(𝛽+1))^{𝑎−𝑝+1}𝑦^𝑐((𝑝−1)𝛼^𝑝−1(𝛼+1))^𝑐𝑥^{𝑎−𝑝+1})^{1/((𝑎−𝑝+1)(𝑒−𝑝+1)−𝑏𝑐)}, (1.7)

𝑛_𝑥0 stands for the outward unit normal at 𝑥₀∈𝜕Ω.

The paper is organized as follows. In Section 2 we consider some preliminaries which will be used in proof of Theorem 1.1. In Section 3 we will give the proof of the main theorem.

By modifications of the arguments in the proof of Theorem 1.1 in [12], we obtain the following main results.

Theorem 1.1. Assume that Ω is a bounded 𝐶² domain of 𝐑^𝐍, 𝑎(𝑥),𝑏(𝑥)∈𝐶^𝜃(Ω) for some 𝜃∈(0,1),𝑎(𝑥),𝑏(𝑥)>0𝑖𝑛Ω and verify (1.2), (𝑎−𝑝+1)(𝑒−𝑝+1)>𝑏𝑐,𝑎,𝑒>𝑝−1,𝑏,𝑐>0,𝛾(𝑥),𝜂(𝑥)∈𝐶(Ω,𝑅⁺) and satisfy 𝑏<𝑒−𝑝+1𝑝+𝛾(𝑥₀)𝑝+𝜂(𝑥₀)<𝑎−𝑝+1𝑐for𝑥₀∈𝜕Ω. (1.8) Then problem (1.1) has a solution (𝑢,𝑣) if and only if 𝑏<𝑒−𝑝+1𝑝+𝛾₁𝑝+𝜂₂,𝑝+𝛾₂𝑝+𝜂₁<𝑎−𝑝+1𝑐. (1.9) And one has lim_𝑥→𝑥0𝑢(𝑥)𝑑(𝑥)^{−𝛼(𝛾(𝑥0),𝜂(𝑥0))}𝐸(𝐷(𝑥₀),𝐶(𝑥₀))=1,lim_𝑥→𝑥0𝑣(𝑥)𝑑(𝑥)^{−𝛽(𝛾(𝑥0),𝜂(𝑥0))}𝐹(𝐷(𝑥₀),𝐶(𝑥₀))=1. (1.10)(1.11)

2. Preliminaries

In this section, we will introduce some propositions.

Definition 2.1. (𝑢,𝑣) is a subsolution of Δ_𝑝𝑢=𝑎(𝑥)𝑢^𝑎𝑣^𝑏ΔinΩ,_𝑝𝑣=𝑏(𝑥)𝑢^𝑐𝑣^𝑒ΔinΩ,provided^𝑝𝑢≥𝑎(𝑥)𝑢^𝑎𝑣^𝑏ΔinΩ,_𝑝𝑣≤𝑏(𝑥)𝑢^𝑐𝑣^𝑒inΩ. (2.1) A supersolution (𝑢,𝑣) is defined by reversing the inequalities.

Proposition 2.2. Assume that (𝑢,𝑣) is a subsolution and (𝑢,𝑣) is a supersolution of problem (1.1), with 𝑢=𝑣=𝑢=𝑣=+∞on𝜕Ω. Then problem (1.1) has at least a solution (𝑢,𝑣) with 𝑢≤𝑢≤𝑢,𝑣≥𝑣≥𝑣inΩ. In particular 𝑢=𝑣=+∞on𝜕Ω.

Proposition 2.3 (see [43]). Assume that 𝑎(𝑥),𝑏(𝑥) satisfy (1.4), then problem (1.1) admits a positive solution (𝑢,𝑣) with 𝑢=𝑣=+∞on𝜕Ω if and only if 𝑘₁,𝑘₂>−𝑝 and 𝑏<𝑒−𝑝+1𝑝+𝑘₁𝑝+𝑘₂<𝑎−𝑝+1𝑐. (2.2) This solution is unique and satisfies lim_𝑥→𝑥0𝑢(𝑥)𝑑(𝑥)^{−𝛼(𝑘1,𝑘2)}𝐸(𝐶₂,𝐶₁)=1,lim_𝑥→𝑥0𝑣(𝑥)𝑑(𝑥)^{−𝛽(𝑘1,𝑘2)}𝐹(𝐶₂,𝐶₁)=1 (2.3) for each 𝑥₀∈𝜕Ω.

Next, we are ready to study two auxiliary problems in a ball and an annuli. To this aim, for given 0<𝑅₁<𝑅 and 𝑥₀∈𝐑^𝐍,𝑁≥1, set

𝐵_𝑅(𝑥₀)={𝑥∈𝐑^𝐍∶|𝑥−𝑥₀|<𝑅},𝐴_𝑅1,𝑅(𝑥₀)={𝑥∈𝐑^𝐍∶𝑅₁<|𝑥−𝑥₀|<𝑅}. (2.4)

Proposition 2.4. Assume Ω=𝐵_𝑅(𝑥₀),(𝑎−𝑝+1)(𝑒−𝑝+1)>𝑏𝑐,𝑎,𝑒>𝑝−1,𝑏,𝑐>0,𝐶(𝑟),𝐷(𝑟)∈𝐶([0,𝑅],𝑅⁺) and 𝛾,𝜂>0 satisfy 𝑏<𝑒−𝑝+1𝑝+𝛾<𝑝+𝜂𝑎−𝑝+1𝑐. (2.5) Then the following systems Δ_𝑝Φ=𝐶(𝑟)(𝑅−𝑟)^𝛾Φ^𝑎Ψ^𝑏in𝐵_𝑅(𝑥₀Δ),_𝑝Ψ=𝐷(𝑟)(𝑅−𝑟)^𝜂Φ^𝑐Ψ^𝑒in𝐵_𝑅(𝑥₀),Φ=Ψ=+∞on𝜕𝐵_𝑅(𝑥₀) (2.6) possess a unique radially symmetric positive solution (Φ(𝑟),Ψ(𝑟)) satisfying lim_𝑥→𝑥0Φ(𝑟)𝐸(𝐷(𝑅),𝐶(𝑅))(𝑅−𝑟)^{−𝛼(𝛾,𝜂)}=1,lim_𝑥→𝑥0Ψ(𝑟)𝐹(𝐷(𝑅),𝐶(𝑅))(𝑅−𝑟)^{−𝛽(𝛾,𝜂)}=1, (2.7) where 𝑟=|𝑥−𝑥₀|.

Proof. At first, we consider the following systems (|Φ^′|^𝑝−2Φ^′)^′+𝑁−1𝑟(|Φ^′|^𝑝−2Φ^′)=𝐶(𝑟)(𝑅−𝑟)^𝛾Φ^𝑎Ψ^𝑏in(0,𝑅),(|Ψ^′|^𝑝−2Ψ^′)^′+𝑁−1𝑟(|Ψ^′|^𝑝−2Ψ^′)=𝐷(𝑟)(𝑅−𝑟)^𝜂Φ^𝑐Ψ^𝑒in(0,𝑅),Φ(𝑅)=Ψ(𝑅)=+∞,Φ^′(0)=Ψ^′(0)=0. (2.8) We will show that problem (2.8) has a solution (Φ(𝑟),Ψ(𝑟)), which provide a positive radially symmetric solution to problem (2.6). Indeed, any positive solution (Φ(𝑟),Ψ(𝑟)) of the integral equation system Φ(𝑟)=𝑙+^𝑟₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐶(𝑠)(𝑅−𝑠)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑠]^1/(𝑝−1)𝑑𝑡,0<𝑟<𝑅,Ψ(𝑟)=𝑚+^𝑟₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐷(𝑠)(𝑅−𝑠)^𝜂Φ^𝑐Ψ^𝑒𝑑𝑠]^1/(𝑝−1)𝑑𝑡,0<𝑟<𝑅, (2.9) provides a solution of (2.8), where Φ(0)=𝑙,Ψ(0)=𝑚,Φ(𝑅)=+∞,Ψ(𝑅)=+∞.
Define Φ₀(𝑟)=𝑙,Ψ₀(𝑟)=𝑚 for all 0<𝑟<𝑅, let {Φ_𝑘},{Ψ_𝑘} be the function sequences given by Φ_𝑘(𝑟)=𝑙+^𝑟₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐶(𝑠)(𝑅−𝑠)^𝛾Φ^𝑎_𝑘−1Ψ^𝑏_𝑘−1𝑑𝑠]^1/(𝑝−1)Ψ𝑑𝑡,0<𝑟<𝑅,_𝑘(𝑟)=𝑚+^𝑟₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐷(𝑠)(𝑅−𝑠)^𝜂Φ^𝑐_𝑘−1Ψ^𝑒_𝑘−1𝑑𝑠]^1/(𝑝−1)𝑑𝑡,0<𝑟<𝑅, (2.10) subject to Φ_𝑘(0)=𝑙,Ψ_𝑘(0)=𝑚,Φ_𝑘(𝑅)=Ψ_𝑘(𝑅)=𝑘.
We remark that {Φ_𝑘},{Ψ_𝑘} are nondecreasing sequences. In fact, Φ₁(𝑟)=𝑙+(𝑙^𝑎𝑚^𝑏)^{1/(𝑝−1)𝑟}₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐶(𝑠)(𝑅−𝑠)^𝛾𝑑𝑠]^1/(𝑝−1)𝑑𝑡=𝑙+(𝑙^𝑎𝑚^𝑏)^1/(𝑝−1)𝐴(𝑟)≥𝑙=Φ₀Ψ(𝑟),₁(𝑟)=𝑚+(𝑙^𝑐𝑚^𝑒)^{1/(𝑝−1)𝑟}₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐷(𝑠)(𝑅−𝑠)^𝜂𝑑𝑠]^1/(𝑝−1)𝑑𝑡=𝑚+(𝑙^𝑐𝑚^𝑒)^1/(𝑝−1)𝐵(𝑟)≥𝑚=Ψ₀(𝑟), (2.11) where 𝐴(𝑟)=^𝑟₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐶(𝑠)(𝑅−𝑠)^𝛾𝑑𝑠]^1/(𝑝−1)𝑑𝑡,𝐵(𝑟)=^𝑟₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐷(𝑠)(𝑅−𝑠)^𝜂𝑑𝑠]^1/(𝑝−1)𝑑𝑡. (2.12) Proceeding by the same manner, we conclude that 𝑙≤Φ_𝑘≤Φ_𝑘+1,𝑚≤Ψ_𝑘≤Ψ_𝑘+1. (2.13)
We now prove that {Φ_𝑘},{Ψ_𝑘} are bounded in (0,𝑅). To prove this, we consider Δ_𝑝Υ=(𝐶(𝑟)(𝑅−𝑟)^𝛾+𝐷(𝑟)(𝑅−𝑟)^𝜂)(Υ^𝑎+𝑏+Υ^𝑐+𝑒), (2.14) problem (2.14) has a large radially symmetric solution Υ(𝑟), and Υ(𝑟)=Υ(0)+^𝑟₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1(𝐶(𝑠)(𝑅−𝑠)^𝛾+𝐷(𝑠)(𝑅−𝑠)^𝜂)(Υ^𝑎+𝑏+Υ^𝑐+𝑒)𝑑𝑠]^1/(𝑝−1)𝑑𝑡, (2.15) where Υ(0)=𝑙+𝑚. It follows that Φ₁(𝑟)=𝑙+(𝑙^𝑎𝑚^𝑏)^{1/(𝑝−1)𝑟}₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝐶(𝑠)(𝑅−𝑠)^𝛾𝑑𝑠]^1/(𝑝−1)𝑑𝑡≤Υ(0)+^𝑟₀[𝑡^1−𝑁𝑡₀𝑠^𝑁−1(𝐶(𝑠)(𝑅−𝑠)^𝛾+𝐷(𝑠)(𝑅−𝑠)^𝜂)(Υ^𝑎+𝑏+Υ^𝑐+𝑒)𝑑𝑠]^1/(𝑝−1)𝑑𝑡=Υ(𝑟). (2.16) Similarly, we have Ψ₁≤Υ(𝑟).
Arguing as before, we obtain Φ_𝑘≤Υ(𝑟),Ψ_𝑘≤Υ(𝑟). Therefore, we show that {Φ_𝑘},{Ψ_𝑘} are nondecreasing and bounded sequences in (0,𝑅), which implies that the following limit holds (Φ,Ψ)=lim_𝑘→∞(Φ_𝑘,Ψ_𝑘), (2.17) we deduce that (Φ,Ψ) is a positive solution of (2.8). Then (Φ(𝑥),Ψ(𝑥))=(Φ(𝑟),Ψ(𝑟)) is a positive radially symmetric solution to problem (2.6) and Φ(𝑅)=lim_𝑟→𝑅Φ(𝑟)=∞,Ψ(𝑅)=lim_𝑟→𝑅Ψ(𝑟)=∞. (2.18) Secondly, it is clear that 𝐶₁(𝑅−𝑟)^𝛾≤𝐶(𝑟)(𝑅−𝑟)^𝛾≤𝐶₂(𝑅−𝑟)^𝛾,𝐷₁(𝑅−𝑟)^𝜂≤𝐷(𝑟)(𝑅−𝑟)^𝜂≤𝐷₂(𝑅−𝑟)^𝜂. (2.19) By (2.5) and Proposition 2.3, we have 𝐸(𝐷₁,𝐶₂)≤lim_𝑟→𝑅Φ(𝑟)(𝑅−𝑟)^{−𝛼(𝛾,𝜂)}≤𝐸(𝐷₂,𝐶₁),𝐹(𝐷₂,𝐶₁)≤lim_𝑟→𝑅Ψ(𝑟)(𝑅−𝑟)^{−𝛽(𝛾,𝜂)}≤𝐹(𝐷₁,𝐶₂). (2.20) Denote by 𝑙=lim_𝑟→𝑅Φ(𝑟)(𝑅−𝑟)^{−𝛼(𝛾,𝜂)},𝑘=lim_𝑟→𝑅Ψ(𝑟)(𝑅−𝑟)^{−𝛽(𝛾,𝜂)}. (2.21) By using 𝛾+𝑝=(𝑎−𝑝+1)𝛼(𝛾,𝜂)+𝑏𝛽(𝛾,𝜂),𝜂+𝑝=(𝑒−𝑝+1)𝛽(𝛾,𝜂)+𝑐𝛼(𝛾,𝜂) and 𝐿^′𝐻𝑜𝑝̂𝑖𝑡𝑎𝑙 rule, we obtain 𝑙=lim_𝑟→𝑅Φ(0)+∫^𝑟₀[𝑡^1−𝑁∫^𝑡₀𝑠^𝑁−1𝐶(𝑠)(𝑅−𝑠)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑠]^1/(𝑝−1)𝑑𝑡(𝑅−𝑟)^−𝛼=lim_𝑟→𝑅[𝑟^1−𝑁∫^𝑟₀𝑡^𝑁−1𝐶(𝑡)(𝑅−𝑡)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑡]^1/(𝑝−1)𝛼(𝑅−𝑟)^−𝛼−1=[lim_𝑟→𝑅𝑟^1−𝑁∫^𝑟₀𝑡^𝑁−1𝐶(𝑡)(𝑅−𝑡)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑡𝛼(𝑅−𝑟)^{−(𝛼+1)(𝑝−1)}]^1/(𝑝−1)=[lim_𝑟→𝑅(1−𝑁)𝑟^−𝑁∫^𝑟₀𝑡^𝑁−1𝐶(𝑡)(𝑅−𝑡)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑡+𝐶(𝑟)(𝑅−𝑟)^𝛾Φ^𝑎Ψ^𝑏𝛼(𝛼+1)(𝑝−1)(𝑅−𝑟)^{−𝛼𝑝+𝛼−𝑝}]^1/(𝑝−1)=[𝐶(𝑅)𝛼(𝛼+1)(𝑝−1)lim_𝑟→𝑅(𝑅−𝑟)^{𝑎𝛼+𝑏𝛽}Φ^𝑎Ψ^𝑏+1−𝑁𝛼(𝛼+1)(𝑝−1)lim_𝑟→𝑅𝑟^−𝑁∫^𝑟₀𝑡^𝑁−1𝐶(𝑡)(𝑅−𝑡)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑡(𝑅−𝑟)^{−𝛼𝑝+𝛼−𝑝}]^1/(𝑝−1)=[𝐶(𝑅)𝑙𝛼(𝛼+1)(𝑝−1)^𝑎𝑘^𝑏+1−𝑁𝛼(𝛼+1)(𝑝−1)lim_𝑟→𝑅𝑟^−𝑁∫^𝑟₀𝑡^𝑁−1𝐶(𝑡)(𝑅−𝑡)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑡(𝑅−𝑟)^{−𝛼𝑝+𝛼−𝑝}]^1/(𝑝−1). (2.22) We note that 0≤lim_𝑟→𝑅𝑟^−𝑁∫^𝑟₀𝑡^𝑁−1𝐶(𝑡)(𝑅−𝑡)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑡(𝑅−𝑟)^{−𝛼𝑝+𝛼−𝑝}≤lim_𝑟→𝑅𝑟⁻¹∫^𝑟₀𝐶(𝑡)(𝑅−𝑡)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑡(𝑅−𝑟)^{−𝛼𝑝+𝛼−𝑝}=lim_𝑟→𝑅𝐶(𝑟)(𝑅−𝑟)^𝛾Φ^𝑎Ψ^𝑏𝑑𝑡𝑅(−𝛼𝑝+𝛼−𝑝)(𝑅−𝑟)^{−𝛼𝑝+𝛼−𝑝+1}=𝐶(𝑅)𝑅(−𝛼𝑝+𝛼−𝑝)lim_𝑟→𝑅(𝑅−𝑟)^{𝛾+𝛼𝑝−𝛼+𝑝+1}Φ^𝑎Ψ^𝑏=𝐶(𝑅)𝑙𝑅(−𝛼𝑝+𝛼−𝑝)^𝑎𝑘^𝑏lim_𝑟→𝑅(𝑅−𝑟)=0. (2.23) This implies that 𝑙^𝑝−1=𝐶(𝑅)𝑙𝛼(𝛼+1)(𝑝−1)^𝑎𝑘^𝑏. (2.24) Similarly, we obtain 𝑘^𝑝−1=𝐷(𝑅)𝑙𝛽(𝛽+1)(𝑝−1)^𝑐𝑘^𝑒. (2.25) Since 𝛼^𝑝−1(𝛼+1)(𝑝−1)𝐶(𝑅)=𝐸(𝐷(𝑅),𝐶(𝑅))^{𝑎−𝑝+1}𝐹(𝐷(𝑅),𝐶(𝑅))^𝑏,𝛽^𝑝−1(𝛽+1)(𝑝−1)𝐷(𝑅)=𝐸(𝐷(𝑅),𝐶(𝑅))^𝑐𝐹(𝐷(𝑅),𝐶(𝑅))^{𝑒−𝑝+1}. (2.26) If 0<𝛼<1,1<𝑝≤2, then 𝐸^{𝑎−𝑝+1}𝐹^𝑏=𝛼^𝑝−1(𝛼+1)(𝑝−1)≥𝐶(𝑅)𝛼(𝛼+1)(𝑝−1)𝐶(𝑅)=𝑙^{𝑎−𝑝+1}𝑘^𝑏, (2.27) therefore, we get 𝐸≥𝑙,𝐹≥𝑘. If 0<𝛼<1,𝑝>2, we get 𝐸≤𝑙,𝐹≤𝑘. So, when 0<𝛼<1, we get 𝐸=𝑙,𝐹=𝑘.
Similarly, when 𝛼≥1, we also get 𝐸=𝑙,𝐹=𝑘.
By (2.24) and (2.25), we conclude that 𝑙=𝐸(𝐷(𝑅),𝐶(𝑅)),𝑘=𝐹(𝐷(𝑅),𝐶(𝑅)), this completes the proof.

Proposition 2.5. Assume (𝑎−𝑝+1)(𝑒−𝑝+1)>𝑏𝑐,𝑎,𝑒>𝑝−1,𝑏,𝑐>0,𝛾>0,𝜂>0, and 𝑏<𝑒−𝑝+1𝑝+𝛾<𝑝+𝜂𝑎−𝑝+1𝑐, (2.28)𝐷(𝑟),𝐶(𝑟)∈𝐶([𝑅₁,𝑅],𝑅⁺) are the reflection around 𝑅₀=(𝑅₁+𝑅)/2 of some functions 𝐶(𝑟),𝐷(𝑟)∈𝐶([𝑅₀,𝑅],𝑅⁺). Then the following system Δ_𝑝Φ=𝐶(𝑟)𝑑(𝑥)^𝛾Φ^𝑎Ψ^𝑏in𝐴_𝑅1,𝑅(𝑥₀Δ),_𝑝Ψ=𝐷(𝑟)𝑑(𝑥)^𝜂Φ^𝑐Ψ^𝑒in𝐴_𝑅1,𝑅(𝑥₀),Φ=Ψ=+∞on𝜕𝐴_𝑅1,𝑅(𝑥₀) (2.29) has a unique radially symmetric positive solution (Φ(𝑟),Ψ(𝑟)) such that lim_{𝑑(𝑥)→0}Φ(𝑟)𝐸(𝐷(𝑅),𝐶(𝑅))𝑑(𝑥)^{−𝛼(𝛾,𝜂)}=1,lim_{𝑑(𝑥)→0}Ψ(𝑟)𝐹(𝐷(𝑅),𝐶(𝑅))𝑑(𝑥)^{−𝛽(𝛾,𝜂)}=1, (2.30) where 𝑑(𝑥)=𝑑(𝑥,𝜕𝐴_𝑅1,𝑅(𝑥₀))=𝑅−|𝑥−𝑥₀|,if𝑅₀≤|𝑥−𝑥₀|≤𝑅,|𝑥−𝑥₀|−𝑅₁,if𝑅₁≤|𝑥−𝑥₀|≤𝑅₀. (2.31)

Proof. The proof is similarl to the proof of Proposition 2.4, so we omit it here.

3. Proof of Theorem 1.1

We are now ready to prove Theorem 1.1, whose proof will be split into the following several lemmas.

Lemma 3.1. Assume (𝑎−𝑝+1)(𝑒−𝑝+1)>𝑏𝑐,𝑎,𝑒>𝑝−1,𝑏,𝑐>0,𝐶(𝑥),𝐷(𝑥)∈𝐶(Ω),𝑎(𝑥),𝑏(𝑥)>0𝑖𝑛Ω and (1.2) holds, 𝛾(𝑥),𝜂(𝑥)>0 and satisfy 𝑏<𝑒−𝑝+1𝑝+𝛾(𝑥₀)𝑝+𝜂(𝑥₀)<𝑎−𝑝+1𝑐, (3.1) for each 𝑥₀∈𝜕Ω, then problem (1.1) has a solution (𝑢,𝑣) if 𝑏<𝑒−𝑝+1𝑝+𝛾₁𝑝+𝜂₂,𝑝+𝛾₂𝑝+𝜂₁<𝑎−𝑝+1𝑐. (3.2)

Proof. By (3.2) and Proposition 2.3, the following system Δ_𝑝Φ=𝐶₁𝑑(𝑥)^𝛾1Φ^𝑎Ψ^𝑏ΔinΩ,_𝑝Ψ=𝐷₂𝑑(𝑥)^𝜂2Φ^𝑐Ψ^𝑒inΩ,Φ=Ψ=+∞on𝜕Ω (3.3) possesses a positive solution (𝑢₁,𝑣₁).
Next we will show that (𝑢,𝑣)=((𝑚+𝑛(𝑚−𝑛)^{𝑏/(𝑎−𝑝+1)})^1/(𝑝−1)𝑢₁,(𝑚−𝑛(𝑚+𝑛)^{𝑐/(𝑒−𝑝+1)})^1/(𝑝−1)𝑣₁) (3.4) is a supersolution of (1.1), if 𝑚 is sufficiently large and 0<𝑚−𝑛<1, where 𝑚,𝑛∈𝑅⁺ and 𝑚>𝑛. In fact, by 𝛾₁+𝑝=(𝑎−𝑝+1)𝛼(𝛾₁,𝜂₂)+𝑏𝛽(𝛾₁,𝜂₂),𝜂₂+𝑝=(𝑒−𝑝+1)𝛽(𝛾₁,𝜂₂)+𝑐𝛼(𝛾₁,𝜂₂). (3.5) We have (𝑢,𝑣) is a supersolution of (1.1) provided 𝐶₁𝑑(𝑥)^𝛾1≤𝑎(𝑥)(𝑚+𝑛)^{((𝑒−𝑝+1)(𝑎−𝑝+1)−𝑏𝑐)/(𝑒−𝑝+1)(𝑝−1)},𝐷₂𝑑(𝑥)≥𝑏(𝑥)(𝑚−𝑛)^{((𝑎−𝑝+1)(𝑒−𝑝+1)−𝑏𝑐)/(𝑎−𝑝+1)(𝑝−1)}. (3.6) Since 𝑎(𝑥),𝑏(𝑥)∈𝐶(Ω), choosing 𝑚 is large enough, and 𝑚−𝑛>0 is sufficiently small, we can prove that (𝑢,𝑣)=((𝑚−𝑛(𝑚+𝑛)^{𝑏/(𝑎−𝑝+1)})^1/(𝑝−1)𝑢₂,(𝑚+𝑛(𝑚−𝑛)^{𝑐/(𝑒−𝑝+1)})^1/(𝑝−1)𝑣₂) (3.7) is a subsolution of (1.1), where (𝑢₂,𝑣₂) is a solution of the following problem: Δ_𝑝Φ=𝐶₂𝑑(𝑥)^𝛾2Φ^𝑎Ψ^𝑏ΔinΩ,_𝑝Ψ=𝐷₁𝑑(𝑥)^𝜂1Φ^𝑐Ψ^𝑒inΩ,Φ=Ψ=+∞on𝜕Ω. (3.8) Then by Proposition 2.2, problem (1.1) has a solution.

Lemma 3.2. Assume that problem (1.1) has a solution (𝑢,𝑣), then (1.9) holds.

Proof. In fact, if (1.9) does not hold, it will lead to a contradiction. From Lemma 3.1, we find that if 𝑚 is large enough and 𝑚−𝑛>0 is sufficiently small, we have 𝑢≤𝑢=(𝑚+𝑛(𝑚−𝑛)^{𝑏/(𝑎−𝑝+1)})^1/(𝑝−1)𝑢₁,𝑣≤𝑣=(𝑚+𝑛(𝑚−𝑛)^{𝑐/(𝑒−𝑝+1)})^1/(𝑝−1)𝑣₂. (3.9) On the other hand, by (2.3), there exists 𝜀>0 such that for 𝑥∈Ω_𝜀={𝑥∈Ω∶𝑑(𝑥,𝜕Ω)≤𝜀}, we get 𝑢≤(𝑚+𝑛(𝑚−𝑛)^{𝑏/(𝑎−𝑝+1)})^1/(𝑝−1)𝑢₁≤(𝑚+𝑛(𝑚−𝑛)^{𝑏/(𝑎−𝑝+1)})^1/(𝑝−1)𝐸(𝐷₂,𝐶₁)𝑑(𝑥)^{−𝛼(𝛾1,𝜂2)}. (3.10) Thus, if 𝑏≥𝑒−𝑝+1𝑝+𝛾₁𝑝+𝜂₂, (3.11) by the definition of 𝛼(𝛾₁,𝜂₂), we obtain 𝛼(𝛾₁,𝜂₂)≤0. By (3.10), it implies that 𝑢 is bounded for 𝑥∈Ω_𝜀, which is impossible since 𝑢(𝑥)=+∞ as 𝑑(𝑥)=dist(𝑥,𝜕Ω)→0⁺. If 𝑝+𝛾₂𝑝+𝜂₂≥𝑎−𝑝+1𝑐, (3.12) it is similarly proved that 𝑣 is bounded near 𝜕Ω, which is also a contradiction. The proof of Lemma 3.2 is complete.

Lemma 3.3. Let (𝑢,𝑣) be a positive solution of (1.1), then (1.10) and (1.11) hold.

Proof. Fix 𝜏∈(0,1), by (1.2), there exits 𝜎∈(0,1) such that, if 𝑑(𝑥,𝑥₀)<𝜎, 𝑎(𝑥)≥(1−𝜏)𝐶(𝑥₀)𝑑(𝑥)^𝛾(𝑥0),𝑏(𝑥)≤(1+𝜏)𝐷(𝑥₀)𝑑(𝑥)^𝜂(𝑥0), (3.13) where 𝑥₀∈𝜕Ω. For a fixed 𝑥₀∈𝜕Ω, set Σ=𝐵_𝜎/2𝜕Ω (3.14) and choose 𝑅>0 small enough such that 𝐾=_𝑦∈Σ𝐵_𝑅(𝑦−𝑅𝑛_𝑦)⊂𝐵_𝜎(𝑥₀)Ω, (3.15) where 𝑛_𝑦 stands for the outward unit normal at 𝑦∈𝜕Ω.
For 𝑥∈𝐵_𝜎(𝑥₀)∩Ω, we get 𝑎(𝑥)≥(1−𝜏)𝐶(𝑥₀)𝑑(𝑥)^𝛾(𝑥0),𝑏(𝑥)≤(1+𝜏)𝐷(𝑥₀)𝑑(𝑥)^𝜂(𝑥0). (3.16) Since Ω is of 𝐶² bounded domain, there exit 𝑅>0 and 𝜎₀>0 such that 𝐵_𝑅(𝑥₀−(𝑅+𝜎)𝑛_𝑥0)⊂Ω,𝐵_𝑅(𝑥₀−𝑅𝑛_𝑥0)𝜕Ω={𝑥₀}, (3.17) for each 𝜎∈(0,𝜎₀).
Let (𝑢_𝐵,𝜎,𝑣_𝐵,𝜎) be any positive radially symmetric solution to the following system: Δ_𝑝𝑢=(1−𝜏)𝐶(𝑥₀)(𝑅−|𝑥−𝑥₀|)^𝛾(𝑥0)𝑢^𝑎𝑣^𝑏in𝐵_𝑅(𝑥₀−(𝑅+𝜎)𝑛_𝑥0Δ),_𝑝𝑣=(1+𝜏)𝐷(𝑥₀)(𝑅−|𝑥−𝑥₀|)^𝜂(𝑥0)𝑢^𝑐𝑣^𝑒in𝐵_𝑅(𝑥₀−(𝑅+𝜎)𝑛_𝑥0),𝑢=𝑣=+∞on𝜕𝐵_𝑅(𝑥₀−(𝑅+𝜎)𝑛_𝑥0). (3.18) It is easy to see that (𝑢_𝜎,𝑣_𝜎)=(𝑢,𝑣)|_{𝐵𝑅(𝑥0−(𝑅+𝜎)𝑛𝑥0)} is a positive smooth subsolution of (3.18), where (𝑢,𝑣) is a positive solution of (1.1).
Then we get 𝑢_𝜎=𝑢|_{𝐵𝑅(𝑥0−(𝑅+𝜎)𝑛𝑥0)}≤𝑢_𝐵,𝜎,𝑣_𝜎=𝑣|_{𝐵𝑅(𝑥0−(𝑅+𝜎)𝑛𝑥0)}≥𝑣_𝐵,𝜎. (3.19)
Let (𝑢_𝐵,𝑣_𝐵) be any positive solution to the following system: Δ_𝑝𝑢=(1−𝜏)𝐶(𝑥₀)(𝑅−|𝑥−𝑥₀|)^𝛾(𝑥0)𝑢^𝑎𝑣^𝑏in𝐵_𝑅(𝑥₀−𝑅𝑛_𝑥0Δ),_𝑝𝑣=(1+𝜏)𝐷(𝑥₀)(𝑅−|𝑥−𝑥₀|)^𝜂(𝑥0)𝑢^𝑐𝑣^𝑒in𝐵_𝑅(𝑥₀−𝑅𝑛_𝑥0),𝑢=𝑣=+∞on𝜕𝐵_𝑅(𝑥₀−𝑅𝑛_𝑥0). (3.20) By Proposition 2.3, (𝑢_𝐵,𝑣_𝐵) satisfies lim_𝑟→𝑅𝑢_𝐵𝐸((1+𝜏)𝐷(𝑥₀),(1−𝜏)𝐶(𝑥₀))(𝑅−𝑟)^{−𝛼(𝛾(𝑥0),𝜂(𝑥0))}=1,lim_𝑟→𝑅𝑣_𝐵𝐹((1+𝜏)𝐷(𝑥₀),(1−𝜏)𝐶(𝑥₀))(𝑅−𝑟)^{−𝛽(𝛾(𝑥0),𝜂(𝑥0))}=1, (3.21)(3.22) where 𝑟=|𝑥−𝑥₀|.
Taking into account that, for 𝑥∈𝐵_𝑅(𝑥₀−(𝑅+𝜎)𝑛_𝑥0), 𝑢_𝐵,𝜎(𝑥)=𝑢_𝐵(𝑥+𝜎𝑛_𝑥0),𝑣_𝐵,𝜎(𝑥)=𝑣_𝐵(𝑥+𝜎𝑛_𝑥0), (3.23) by (3.19), for each 𝑥∈𝐵_𝑅(𝑥₀−(𝑅+𝜎)𝑛_𝑥0) and 𝜎∈(0,𝜎₀), we have 𝑢(𝑥)≤𝑢_𝐵(𝑥+𝜎𝑛_𝑥0),𝑣(𝑥)≥𝑣_𝐵(𝑥+𝜎𝑛_𝑥0). (3.24) Let 𝜎→0, we have 𝑢(𝑥)≤𝑢_𝐵(𝑥),𝑣(𝑥)≥𝑣_𝐵(𝑥). (3.25) It follows immediately from (3.21), (3.22) that lim_𝑟→𝑅𝑢𝐸(𝑅−𝑟)^−𝛼≤lim_𝑟→𝑅𝑢_𝐵𝐸(𝑅−𝑟)^−𝛼=1,lim_𝑟→𝑅𝑣𝐹(𝑅−𝑟)^−𝛽≥lim_𝑟→𝑅𝑣_𝐵𝐹(𝑅−𝑟)^−𝛽=1, (3.26)(3.27) where 𝐸=𝐸((1+𝜏)𝐷(𝑅),(1−𝜏)𝐶(𝑅)),𝐹=𝐹((1+𝜏)𝐷(𝑅),(1−𝜏)𝐶(𝑅)).
We next have to prove the inverse inequalities. Similarly, there exits 𝑅>𝑅₁>0 and 𝜎₀>0 such that Ω⊂⋂_0<𝜎<𝜎0𝐴_𝑅1,𝑅(𝑥₀+(𝑅+𝜎)𝑛_𝑥0) and 𝐴_𝑅0,𝑅(𝑥₀+𝑅₁𝑛_𝑥0)⋂𝜕Ω={𝑥₀}. Fix a sufficiently small 𝜏, there exit radially symmetric functions 𝑎∶𝐴_𝑅1,𝑅(𝑥₀+𝑅₁𝑛_𝑥0)→𝑅⁺ and 𝑏∶𝐴_𝑅1,𝑅(𝑥₀+𝑅₁𝑛_𝑥0)→𝑅⁺ such that 𝑎≥𝑎,𝑏≤𝑏 in Ω, and max_{𝐴𝑅101𝑛𝑥0),𝑅(𝑥+𝑅}𝑎≤max_Ω𝑎+1,max_Ω𝑏+1≤max_{𝐴𝑅101𝑛𝑥0),𝑅(𝑥+𝑅}𝑏, (3.28) and for each 𝑥∈𝐴_𝑅1,𝑅(𝑥₀+𝑅₁𝑛𝑥₀)𝑎(𝑥)=𝑎₁(|𝑥−𝑥₀−𝑅₁𝑛_𝑥0|)[𝑑(𝑥,𝜕𝐴_{𝑅1,𝑅(𝑥0+𝑅1𝑛𝑥0)})]^𝛾(𝑥0),𝑏(𝑥)=𝑏₁(|𝑥−𝑥₀−𝑅₁𝑛_𝑥0|)[𝑑(𝑥,𝜕𝐴_{𝑅1,𝑅(𝑥0+𝑅1𝑛𝑥0)})]^𝜂(𝑥0), (3.29) where 𝑎₁,𝑏₁∈𝐶([𝑅₁,𝑅],𝑅⁺), satisfing 𝑎₁(𝑅₁)=𝐶(𝑥₀)+𝜏,𝑏₁(𝑅₁)=𝐷(𝑥₀)−𝜏. (3.30)
We now consider the system Δ_𝑝𝑢=𝑎(𝑥)𝑢^𝑎𝑣^𝑏in𝐴_𝑅1,𝑅(𝑥₀+𝑅₁𝑛_𝑥0Δ),_𝑝𝑣=𝑏(𝑥)𝑢^𝑐𝑣^𝑒in𝐴_𝑅1,𝑅(𝑥₀+𝑅₁𝑛_𝑥0),𝑢=𝑣=+∞on𝜕𝐴_𝑅1,𝑅(𝑥₀+𝑅₁𝑛_𝑥0). (3.31) By Proposition 2.5, problem (3.31) possesses a solution (𝑢_𝐴,𝑣_𝐴).
But for the system Δ_𝑝𝑢=𝑎(𝑥)𝑢^𝑎𝑣^𝑏in𝐴_𝑅1,𝑅(𝑥₀+(𝑅₁+𝜎)𝑛_𝑥0Δ),_𝑝𝑣=𝑏(𝑥)𝑢^𝑐𝑣^𝑒in𝐴_𝑅1,𝑅(𝑥₀+(𝑅₁+𝜎)𝑛_𝑥0),𝑢=𝑣=+∞on𝜕𝐴_𝑅1,𝑅(𝑥₀+(𝑅₁+𝜎)𝑛_𝑥0), (3.32) it has a solution (𝑢_𝐴,𝜎,𝑣_𝐴,𝜎), and for each 𝑥∈𝐴_𝑅1,𝑅(𝑥₀+(𝑅₁+𝜎)𝑛_𝑥0), we have (𝑢_𝐴,𝜎(𝑥),𝑣_𝐴,𝜎(𝑥))=(𝑢_𝐴(𝑥−𝜎𝑛_𝑥0),𝑣_𝐴(𝑥−𝜎𝑛_𝑥0)). (3.33) It is also clear that (𝑢_𝐴(𝑥),𝑣_𝐴(𝑥))=(𝑢_𝐴,𝜎(𝑥),𝑣_𝐴,𝜎(𝑥))|_Ω is a subsolution of problem (1.1). Thus for each 𝑥∈𝐴_𝑅1,𝑅(𝑥₀+(𝑅₁+𝜎)𝑛_𝑥0), we get 𝑢_𝐴(𝑥−𝜎𝑛_𝑥0)≤𝑢(𝑥),𝑣_𝐴(𝑥−𝜎𝑛_𝑥0)≥𝑣(𝑥). Let 𝜎→0, we have 𝑢_𝐴(𝑥)≤𝑢(𝑥),𝑣_𝐴(𝑥)≥𝑣(𝑥). Thus for 𝑥∈𝐾, we get 1=lim_{|𝑥|→𝑅}𝑢_𝐴(𝑥)𝐸(𝑎(𝑥),𝑏(𝑥))(𝑅−|𝑥|)^{−𝛼(𝛾(𝑥0),𝜂(𝑥0))}≤lim_{𝑑(𝑥)→0}𝑢(𝑥)𝐸(𝑎(𝑥),𝑏(𝑥))(𝑅−|𝑥|)^{−𝛼(𝛾(𝑥0),𝜂(𝑥0))},1=lim_{|𝑥|→𝑅}𝑣_𝐴(𝑥)𝐹(𝑎(𝑥),𝑏(𝑥))(𝑅−|𝑥|)^{−𝛽(𝛾(𝑥0),𝜂(𝑥0))}≥lim_{𝑑(𝑥)→0}𝑣(𝑥)𝐹(𝑎(𝑥),𝑏(𝑥))(𝑅−|𝑥|)^{−𝛽(𝛾(𝑥0),𝜂(𝑥0))} (3.34)(3.35) but we have lim_𝜏→0𝐾={𝑥₀}. Therefore, by (3.26), (3.27), (3.34), and (3.35), we finish (1.10) and (1.11). The proof of Lemma 3.3 is complete. From Lemma 3.1 to Lemma 3.3, we finish the proof of Theorem 1.1.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (Grant no. 10871060) by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant no. 8KJB110005).