Abstract and Applied Analysis

VolumeΒ 2010, Article IDΒ 971268, 20 pages

http://dx.doi.org/10.1155/2010/971268

## Existence and Asymptotic Behavior of Boundary Blow-Up Solutions for Weighted -Laplacian Equations with Exponential Nonlinearities

^{1}Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China^{2}Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China^{3}School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, Henan 454000, China^{4}School of Mathematics and Statistics, Huazhong Normal University, Wuhan, Hubei 430079, China

Received 21 March 2010; Accepted 3 October 2010

Academic Editor: PavelΒ DrΓ‘bek

Copyright Β© 2010 Li Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the following -Laplacian equations with exponential nonlinearities: in , as , where is called -Laplacian, . The asymptotic behavior 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. On the background of this class of problems, we refer to [1β3]. Many results have been obtained on this kind of problems, for example, [4β18]. On the regularity of weak solutions for differential equations with nonstandard -growth conditions, we refer to [4, 5, 8]. On the existence of solutions for -Laplacian equation Dirichlet problems in bounded domain, we refer to [7, 9, 15, 18]. In this paper, we consider the following -Laplacian equations with exponential nonlinearities

where and is a bounded radial domain (). Our aim is to give the asymptotic behavior and the existence of boundary blow-up solutions for problem (P).

Throughout the paper, we assume that , , and satisfy the following.

(H_{1}) is radial and satisfies

(H_{2}) is radial with respect to , is increasing, and for any .

(H_{3})β is continuous and satisfies

where , are positive constants and .

(H_{4}) is a radial nonnegative function, and there exists a constant such that

where and are positive constants and and are Lipschitz continuous on , which satisfy for any .

The operator is called -Laplacian. Specifically, if (a constant), (P) is the well-known -Laplacian problem. If can be represented as , on the boundary blow-up solutions for the following -Laplacian equations ( is a constant):

we refer to [19β26], and the following generalized Keller-Osserman condition is crucial

but the typical form of -Laplacian equation is

and there are some differences between the results of (1.4) and (1.6) (see [16]).

On the boundary blow-up solutions for the following -Laplacian equations with exponential nonlinearities ( is a constant):

we refer to [20β22], but the results on the boundary blow-up solutions for -Laplacian equations are rare (see [16]).

In [16], the present author discussed the existence and asymptotic behavior of boundary blow-up solutions for the following -Laplacian equations:

on the condition that satisfies polynomial growth condition.

If is a function, the typical form of (P) is the following:

and the method to construct subsolution and supersolution in [16] cannot give the exact asymptotic behavior of solutions for (P). Our results partially generalized the results of [20β22].

Because of the nonhomogeneity of -Laplacian, -Laplacian problems are more complicated than those of -Laplacian ones (see [10]); 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 the spaces , and properties of -Laplacian, which we will use later (see [6, 11]). Let

We can introduce the norm on by

The space (, ) becomes a Banach space. We call it generalized Lebesgue space. The space (, ) is a separable, reflexive, and uniform convex Banach space (see [6, Theorems 1.10, 1.14] ).

The space is defined by

and it can be equipped with the norm

is the closure of in . and are separable, reflexive, and uniform convex Banach spaces (see [6, Theorem 2.1]).

If , is called a blow-up solution of (P) when it satisfies

for any domain , and for every positive integer .

Let there is an open domain such that , and define as

Lemma 2.1 (see [9, Theorem 3.1]). *Let , and . Then, is strictly monotone.*

Letting , if , with 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 [14], we have the following.

Lemma 2.3 (comparison principle). *Let satisfy
**
Let . If (i.e., on ), then a.e. in .*

Lemma 2.4 (see [8, Theorem 1.1]). *Under the conditions (H) and (H), if is a bounded weak solution of in , then , where is a constant.*

#### 3. Asymptotic Behavior of Boundary Blow-Up Solutions

If is a radial solution for (P), then (P) can be transformed into

It means that is increasing.

Theorem 3.1. *If satisfies
**
where is defined in (H) and Ξ± and are positive constants, then there exists a supersolution which satisfies (as ), such that for every solution of problem (P), one has .*

*Proof. *Define the function on as
where , are constants, , is small enough, parameter , satisfies ,

Obviously, for any positive constant , we have .

When , we have

where

If is small enough, it is easy to see that

and then

Thus, when is small enough, from (3.5) and (3.8), for uniformly, we have

Thus, when is small enough, the following inequality is valid for uniformly:

Obviously, if is small enough, then is large enough. Since ,

Thus,

Obviously,

Since is a function on , if is small enough ( depends on , , , *Ξ±*), from (3.10), (3.12), and (3.13), for any , we can see that is a supersolution for (P) on , and then is a supersolution for (P).

Defining the function on , where , then is a supersolution for (P) on . If is a solution for (P), according to the comparison principle, we get that for any . For any , we have , when is large enough. Thus

When is small enough, we have

According to the comparison principle, we get that ; then is a radial upper control function of all of the solutions for (P), and is a radial supersolution for (P). The proof is completed.

Theorem 3.2. *If satisfies
**
where is defined in (H _{4}) and and are positive constants, then there exists a subsolution which satisfies (as ), such that for every solution for problem (P), one has .*

*Proof. *We will prove this theorem in the following two cases.(i). (ii). *Case 1 (). *Let be a radial solution of

where is a positive constant. We denote . Then, satisfies

Denote on as

It is easy to see that
Define the function on as
where , are constants, , is small enough, parameter , and
where satisfies

Obviously, for any positive constant , .

By computation, when , we have

where

By computation, when is small enough, for uniformly, we have

Then, for uniformly, we have

When is small enough, , since , it is easy to see that

Then,

Obviously,

Combining (3.27), (3.29), and (3.30), when is large enough, for any , one can see that is a subsolution for (P).

Define the function on as

where is a small enough positive constant such that .

For any , we can see that is a subsolution for (P) on . According to the comparison principle, we get that for any . For any , we have . Thus

When is small enough, we have.

According to the comparison principle, we get that ; then is a radial lower control function of all of the solutions for (P), and is a radial subsolution for (P).*Case 2 (). *Let be a positive constant. Denote on as

It is easy to see that

Define the function on as
where is a constant, , is small enough, parameter , and
where is defined in (3.23).

Obviously, for any positive constant , .

Similar to the proof of Case 1, when is small enough, we have

When is small enough, , from the definition of , it is easy to see that

Obviously

Combining (3.37), (3.38), and (3.39), when is large enough, for any , one can see that is a subsolution for (P).

Define the function on as

where is a small enough positive constant such that .

We can see that is a subsolution for (P) for any . According to the comparison principle, we get that for any . For any , we have . Then,

When is small enough, we have

According to the comparison principle, we get that ; then is a radial lower control function of all of the solutions for (P), and is a radial subsolution for (P).

Theorem 3.3. *If satisfies
**
where is defined in (H _{4}), and are positive constants, , where , then each solution for (P) satisfies
*

*Proof. *It is easy to be seen from Theorems 3.1 and 3.2

#### 4. The Existence of Boundary Blow-Up Solutions

Theorem 4.1. *If and satisfies
**
where is defined in (H _{4}), and are positive constants, then (P) possesses a boundary blow-up solution.*

*Proof. *In order to deal with the existence of boundary blow-up solutions, let us consider the problem
where is a positive constant and is a radial subdomain of . Since , then , where . The relative functional of (4.2) is
where . Since is coercive in , then possesses a nontrivial minimum point . So, problem (4.2) possesses a weak solution .

Since , from Theorems 3.1 and 3.2, we get that (P) possesses a supersolution and a subsolution , which satisfy , when (the distance from to ) is small enough. According to the comparison principle, we get that for any .

Denote (). Let us consider the problem

and the relative functional is

Let . Since the functional is coercive in , then has a nontrivial minimum point . Therefore, problem (4.4) has a weak solution .

According to the comparison principle, we get that for any ( ). Since for any , then for any (). According to the comparison principle, we get that for any ().

Since is a supersolution and for any , so we have for any (). According to the comparison principle, we get that for any ().

Since and are locally bounded, from Lemma 2.4, each weak solution of (4.4) is a function. The interior regularity result implies that the sequences and are equicontinuous in , and hence we can choose a subsequence, which we denoted by , such that and uniformly on for some and . In fact, on , and from the interior estimate, we conclude that for some . Thus, . From the interior regularity result, we see that on , and since the function is continuous on , it follows that for . Thus, by the dominated convergence theorem, we have

Furthermore, since and for each , by the monotone convergence theorem, we obtain

Therefore, it follows that

and hence is a weak solution for on .

Thus, there exists a subsequence of which we denote it by, such that in (as ), where and satisfies

Similarly, we can prove that there exists a subsequence of which we denote by, such that in (as ), where satisfies and

Repeating the above steps, we can get a subsequence of which we denote by () and satisfies the following.(1^{0}) For any fixed , is a subsequence of.(2^{0}) For any fixed , in (as ), where satisfies .(3^{0})For any fixed , satisfies

Thus, we can conclude that

(i)is a subsequence of,(ii) there exists a function such that , and for any , there exists a constant such that when , is defined at , and ,(iii)

Obviously, is a boundary blow-up solution for (P).

This completes the proof.

In Theorem 4.1, when , the existence of solutions for (P) is given. In the following, we will consider the existence of solutions for (P) in the general case . We need to do some preparation. Let us consider

where and is a constant.

Lemma 4.2. *If , where and are defined in Theorems 3.13.2, respectively, then (4.13) has a solution satisfying
*

*Proof. *Denote

Let . Let us consider the even solutions of the following

It is easy to see that is an even solution for (4.15) if and only if is even and

Denote . Similar to the proof of Lemma 2.3 of [18], for any , it is easy to see that is compact continuous and bounded from