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:

where is a bounded domain of and stands for the -Laplacian operator defined by The exponents verify There exists such that where

We must emphasize that the weight functions are allowed decaying to zero on with arbitrary rate, depending upon the particular point of . The boundary condition is to be understood as . 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 -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 is a characteristic of the medium. Media with are called dilatant fluids and those with are called pseudoplastics. If , they are Newtonian fluids. When , the problem becomes more complicated since certain nice properties inherent to the case seem to be lost or at least difficult to verify. The main differences between and can be founded in [4, 5].

When , system (1.1) becomes for which the existence, uniqueness, and asymptotic behavior of large solutions have been investigated extensively. We list here, for example, [612].

This is a huge amount of literature dealing with single equation with infinite boundary conditions (see, e.g., [1334]). This problem with more general nonlinearies and weight-function has been discussed by many authors recently [3539].

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

as for some positive constants and real numbers . This problem was later studied in [41] with general form, where

for , are positive constants. The author also obtained uniqueness results.

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

where , and is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: or or on , where . Under several hypotheses on the parameters , the author showed the existence of positive solutions and further provided the asymptotic behavior of the solutions near .

When , in [43], problem (1.1) was analyzed with 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 , and 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

stands for the outward unit normal at

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 is a bounded domain of , for some and verify (1.2), and satisfy Then problem (1.1) has a solution if and only if And one has

2. Preliminaries

In this section, we will introduce some propositions.

Definition 2.1. is a subsolution of A supersolution is defined by reversing the inequalities.

Proposition 2.2. Assume that is a subsolution and is a supersolution of problem (1.1), with Then problem (1.1) has at least a solution with In particular

Proposition 2.3 (see [43]). Assume that satisfy (1.4), then problem (1.1) admits a positive solution with if and only if and This solution is unique and satisfies for each

Next, we are ready to study two auxiliary problems in a ball and an annuli. To this aim, for given and set

Proposition 2.4. Assume and satisfy Then the following systems possess a unique radially symmetric positive solution satisfying where

Proof. At first, we consider the following systems We will show that problem (2.8) has a solution which provide a positive radially symmetric solution to problem (2.6). Indeed, any positive solution of the integral equation system provides a solution of (2.8), where
Define for all , let be the function sequences given by subject to
We remark that are nondecreasing sequences. In fact, where Proceeding by the same manner, we conclude that
We now prove that are bounded in . To prove this, we consider problem (2.14) has a large radially symmetric solution , and where . It follows that Similarly, we have .
Arguing as before, we obtain . Therefore, we show that are nondecreasing and bounded sequences in , which implies that the following limit holds we deduce that is a positive solution of (2.8). Then is a positive radially symmetric solution to problem (2.6) and Secondly, it is clear that By (2.5) and Proposition 2.3, we have Denote by By using and rule, we obtain We note that This implies that Similarly, we obtain Since If , then therefore, we get . If , we get . So, when , we get .
Similarly, when , we also get .
By (2.24) and (2.25), we conclude that , this completes the proof.

Proposition 2.5. Assume and are the reflection around of some functions . Then the following system has a unique radially symmetric positive solution such that where

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 and (1.2) holds, and satisfy for each , then problem (1.1) has a solution if

Proof. By (3.2) and Proposition 2.3, the following system possesses a positive solution .
Next we will show that is a supersolution of (1.1), if is sufficiently large and , where and . In fact, by We have is a supersolution of (1.1) provided Since , choosing is large enough, and is sufficiently small, we can prove that is a subsolution of (1.1), where is a solution of the following problem: Then by Proposition 2.2, problem (1.1) has a solution.

Lemma 3.2. Assume that problem (1.1) has a solution , 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 is large enough and is sufficiently small, we have On the other hand, by (2.3), there exists such that for , we get Thus, if by the definition of , we obtain . By (3.10), it implies that is bounded for , which is impossible since as . If it is similarly proved that is bounded near , which is also a contradiction. The proof of Lemma 3.2 is complete.

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

Proof. Fix , by (1.2), there exits such that, if , where . For a fixed , set and choose small enough such that where stands for the outward unit normal at .
For , we get Since is of bounded domain, there exit and such that for each .
Let be any positive radially symmetric solution to the following system: It is easy to see that is a positive smooth subsolution of (3.18), where is a positive solution of (1.1).
Then we get
Let be any positive solution to the following system: By Proposition 2.3, satisfies where .
Taking into account that, for , by (3.19), for each and , we have Let , we have It follows immediately from (3.21), (3.22) that where .
We next have to prove the inverse inequalities. Similarly, there exits and such that and . Fix a sufficiently small , there exit radially symmetric functions and such that in , and and for each where , satisfing
We now consider the system By Proposition 2.5, problem (3.31) possesses a solution .
But for the system it has a solution , and for each , we have It is also clear that is a subsolution of problem (1.1). Thus for each , we get . Let , we have . Thus for , we get but we have . 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.


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).