Qihu Zhang

Abstract

This paper investigates the p(x)-Laplacian equations with exponential nonlinearities −△p(x)u+ef(x,u)=0 in Ω, u(x)→+∞ as d(x,∂Ω)→0, where −△p(x)u=−div(|∇u|p(x)−2∇u) is called p(x)- Laplacian. The singularity of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.

1. Introduction

The study of differential equations and variational problems with nonstandard -growth conditions is a new and interesting topic. We refer to [1, 2], the background of these problems. Many results have been obtained on this kind of problems, for example, [1–15]. In this paper, we consider the -Laplacian equations with exponential nonlinearities (P) where , is a bounded radial domain (. Our aim is to give the existence and asymptotic behavior of solutions for problem (P).

Throughout the paper, we assume that and satisfy that (H₁) is radial and satisfies (1.1)(H₂) is radial with respect to , is increasing and for any ;(H₃) is a continuous function and satisfies (1.2) where are positive constants, .

The operator is called -Laplacian. Especially, if (a constant), (P) is the well-known -Laplacian problem (see [16–18]).

Because of the nonhomogeneity of -Laplacian, -Laplacian problems are more complicated than those of -Laplacian ones (see [6]); and another difficulty of this paper is that cannot be represented as .

2. Preliminary

In order to deal with -Laplacian problems, we need some theories on spaces and and properties of -Laplacian, which we will use later (see [3, 7]). Let (2.1) We can introduce the norm on by (2.2) The space (, ) becomes a Banach space. We call it generalized Lebesgue space. The space (, ) is a separable, reflexive, and uniform convex Banach space (see [3, Theorems 1.10, 1.14]).

The space is defined by (2.3) and it can be equipped with the norm (2.4) is the closure of in . and are separable, reflexive, and uniform convex Banach spaces (see [3, Theorem 2.1]).

If , is called a solution of (P) if it satisfies (2.5) for any domain , and for any .

Let there exists an open domain s.t. . For any and , define as

Lemma 2.1 (See [5, Theorem 3.1]). Let , Then, is strictly monotone.

Let if for all a.e. in , then denote in correspondingly, if in then denote in .

Definition 2.2. Let If in then is called a weak supersolution (weak subsolution) of (P).

Copying the proof of [9], we have the following lemma.

Lemma 2.3 (Comparison Principle). Let satisfy in . Let . If (i.e., on ), then a.e. in

Lemma 2.4 (See [4, Theorem 1.1]). Under the conditions (H₁) and (H₃), if is a bounded weak solution of in , then where is a constant.

3. Main Results and Proofs

If is a radial solution of (P), then (P) can be transformed into (3.1) It means that is increasing.

Theorem 3.1. If there exists a constant such that (3.2) where and are positive constants, then there exists a continuous function which satisfies (as ), and such that, if is a weak solution of problem (P), then .

Proof. Let . Denote (3.3) Define the function on as (3.4) where is a constant, , and is small enough, and .

Obviously, for any positive constant , .

When , we have (3.5) where (3.6) If is small enough, it is easy to see ; from (3.5), we have (3.7) Obviously, if is small enough, then is large enough, so we have (3.8) Obviously, (3.9) Since is a function on , if is small enough ( depends on ), from (3.7), (3.8), and (3.9), we can see that is a supersolution of (P).

Define the function on as (3.10) where is a big-enough integer such that , , , is a positive small constant such that .

Obviously, is a supersolution of (P) on . If is a solution of (P), according to the comparison principle, we get that for any . For any we have Thus, (3.11) When is small enough, we have (3.12) According to the comparison principle, we obtain that , for all , then is an upper control function of all of the solutions of (P). The proof is completed.

Theorem 3.2. If there exists a such that (3.13) where and are positive constants, then there exists a continuous function which satisfies (as ), and such that, if is a solution of problem (P), then .

Proof. Let be a radial solution of (3.14) where is a positive constant. We denote , then satisfies , , and (3.15) Denote on as (3.16) It is easy to see that (3.17) Define the function on as (3.18) where is a constant, , and is small enough, and .

Obviously, for any positive constant , .

Similar to the proof of Theorem 3.1, when is small enough, we have (3.19) When is small enough, for all , since , then (3.20) Obviously, (3.21) Combining (3.19), (3.20), and (3.21), we can see that is a subsolution of (P).

Define the function on as (3.22) where is a small-enough positive constant such that

We can see that is a subsolution of (P) on , according to the comparison principle, we get that for any . For any we have Thus, (3.23) When is small enough, we have (3.24) From the comparison principle, we obtain , then is a lower control function of all of the solutions of (P).

Theorem 3.3. If and there exists a such that (3.25) where and are positive constants, then (P) possesses a solution.

Proof. In order to deal with the existence of boundary blow-up solutions of (P), let us consider the problem (3.26) where . Since , then , where . The relative functional of (3.26) is (3.27) where . Since is coercive in then possesses a nontrivial minimum point , then problem (3.26) possesses a weak solution . According to the comparison principle, we get for any and . Since defined in Theorem 3.1 is a supersolution, according to the comparison principle, we have on for all . Since is locally bounded, from Lemma 2.4, every weak solution of (P) is a locally function. Thus, possesses a subsequence (we still denote it by ), such that is a solution of (P).

Acknowledgments

This work was supported by the National Science Foundation of China (10701066 & 10671084)and China Postdoctoral Science Foundation (20070421107) and the Natural Science Foundation of Henan Education Committee (2007110037).

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