Existence and Nonexistence of Positive Solutions for Singular -Laplacian Equation in

Abstract

We study the existence and nonexistence of solutions for the singular quasilinear problem , , , , , where , and behave like and with at the origin. We obtain the existence by the upper and lower solution method and the nonexistence by the test function method.

1. Introduction

In this paper, we study through the upper and lower solution method and the test function method the existence and nonexistence of solution to the singular quasilinear elliptic problem

(1.1)

with , , . are the locally Hölder continuous functions, not identically zero and ) and are locally Lipschitz continuous functions.

The study of this type of equation in (1.1) is motivated by its various applications, for instance, in fluid mechanics, in Newtonian fluids, in flow through porous media, and in glaciology; see [1]. The equation in (1.1) involves singularities not only in the nonlinearities but also in the differential operator.

Many authors studied this kind of problem for the case ; see [2–7]. In these works, the nonlinearities have sublinear and suplinear growth at infinity, and they behave like a function (, or ) at the origin. Roughly speaking, in this case we say that the nonlinearities are concave and convex or "slow diffusion and fast diffusion''; see [8].

When , , and , , by using the lower and upper solution method, Santos in [5] finds a real number , such that the problem (1.1) has at least one solution if .

For , the existence and multiplicity of solution of singular elliptic equation like (1.1) in a bounded domain with the zero Dirichlet data have been widely studied by many authors, for example, the authors [9–13] and references therein. Assunção et al. in [14] studied the multiplicity of solution for the singular equations in (1.1) with , , , and in . Similar consideration can be found in [15–20] and references therein. We note that the variation method is widely used in the above references.

Recently, Chen et al. in [21, 22], by using a variational approach, got some existence of solution for (1.1) with and , . For the case , , the problem for the existence of solution for (1.1) is still open. It seems difficult to consider the case by variational method.

The main aim of this work is to study the existence and nonexistence of solution for (1.1), where is sublinear and is suplinear. We will use the upper and lower solution method. To the best of our knowledge, there is little information on upper and lower solution method for the problem (1.1). So it is necessary to establish this technique in unbounded domain. To obtain the existence, the assumption (see (2.17) below) is essential. By this, an upper solution for (1.1) is obtained.

We also obtain a sufficient condition on , to guarantee the nonexistence of nontrivial solution for the problem (2.21). (see Theorem 2.5 below). It must be particularly pointed out that our primary interest is in the mixed case in which with satisfying

(1.2)

while satisfies

(1.3)

This paper is organized as follows. In Section 2, we state the main results and present some preliminaries which will be used in what follows. We also introduce the precise hypotheses under which our problem is studied. In Section 3, we give the proof of some lemmas and the existence. The proof of nonexistence is given in Section 4.

2. Preliminaries and Main Results

Let us now introduce some weighted Sobolev spaces and their norms. Let be a bounded domain in with smooth boundary . If and , we define as being the subspace of of the Lebesgue measurable function , satisfying

(2.1)

If and , we define (resp., as being the closure of (resp., ) with respect to the norm defined by

(2.2)

For the weighted Sobolev space , we have the following compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact theorem.

Theorem 2.1 ((compact imbedding theorem) [13]).

Suppose that is an open bounded domain with boundary and , , . Then, the imbedding is compact if , .

We now consider the existence of positive solutions for problem (1.1). Our main tool will be the upper and lower solution method. This method, in the bounded domain situation, has been used by many authors, for instance, [10, 12, 13]. But for the unbounded domain, we need to establish this method and then to construct an upper solution and a lower solution for (1.1). We now give the definitions of upper and lower solutions.

Definition 2.2 (see [10, 12]).

A function is said to be a weak lower solution of the equation

(2.3)

if

(2.4)

or

(2.5)

for any , .

Similarly, a function is said to be a weak upper solution of (2.3) if

(2.6)

or

(2.7)

for any and in .

A function is said to be a weak solution of (2.3) if and only if is a weak lower solution and weak upper solution of (2.3).

A function is said to be less than or equal to on if .

If and , we define the weighted Sobolev space as being the closure of with respect to the norm defined by

(2.8)

The following lemma will be basic in our approach.

Lemma 2.3.

Let be Lipschitz continuous and nondecreasing in and locally Hölder continuous in . Moreover, assume that there exist the functions such that

(2.9)

Then, there exist a minimal weak solution and a maximal weak solution of (2.3) satisfying

(2.10)

and .

Proof.

Denote , . Let be a pair of upper and lower solutions of (2.3) with , a.e. in . We consider the boundary value problem

(2.11)

By Theorem  1.1 in [10], one concludes that there exists which is a weak solution of (2.11) with a.e. in for .

We define its extension by

(2.12)

Similarly, let be a weak solution of the boundary value problem

(2.13)

and its extension is defined by

(2.14)

Since , we have . By Theorem  2.4 in [12], we have

(2.15)

for . In view of (2.15), the pointwise limits

(2.16)

exist and in .

Similar to the proof Theorem  1.1 in [10] and the proof of Theorem  7.5.1 in [23], it is not difficult to get from Theorem 2.1 that is the maximal weak solution and the minimal solution of (2.3), which satisfies (2.10) and . This ends the proof of Lemma 2.3.

Our main results read as follows.

Theorem 2.4 (existence).

Let , . Assume the following.

The nonnegative functions are Lipschitz continuous and nondecreasing, . Additionally, and with .

The nonnegative functions , are locally Hölder continuous. Let . If

(2.17)

then there exists , such that , and the problem (1.1) admits a weak solution .

Theorem 2.5 (nonexistence).

Let , . Assume that

;

there exist such that

(2.18)

the functions in satisfy

(2.19)

where and

(2.20)

Then the problem

(2.21)

has no nontrivial solution .

Remark 2.6.

If assumption (2.19) holds, then

(2.22)

with , .

In fact, for this case, there exist and such that

(2.23)

for . Therefore,

(2.24)

So, condition (2.19) implies (2.22).

3. Proof of Existence

Before proofing the existence, we present some preliminary lemmas which will be useful in what follows.

Lemma 3.1.

Suppose that , is local Hölder continuous and satisfies

(3.1)

Then the problem

(3.2)

has a weak solution , where .

Proof.

Let . Then . Denote

(3.3)

Obviously, and . It is easy to verify that

(3.4)

This shows that (resp., ) is a lower (resp., upper) solution of (3.2). Then by Lemma 2.3, there exists a weak solution for problem (3.2) satisfying , and

(3.5)

Lemma 3.2.

Let . If

• (1)
  

  (3.6)

• (2)
  

  (3.7)

one has .

Proof.

• (1) Since , . By the Hölder inequality, we obtain

  

  (3.8)

• (2) If and , we take and then .

Note that

(3.9)

This implies and ends the proof of Lemma 3.2.

Corollary 3.3.

If satisfies the conditions in Lemma 3.2, then the problem (3.2) admits a solution .

Lemma 3.4.

Suppose that is nondecreasing and with . Additionally, let the function be locally Hölder continuous and satisfy

(3.10)

where . Then the problem

(3.11)

has a weak solution .

Proof.

We first consider the problem

(3.12)

By Lemma 3.1, there is a solution for (3.12) satisfying . In order to get the existence of solution for (3.11), we chose a pair of upper-lower solution of the equation in (3.11) by means of .

Let . It is easy to verify that is an upper solution of

(3.13)

if and only if

(3.14)

or

(3.15)

By the assumption on , we know that there exists , such that . So, . Then we take so that is an upper solution of (3.13).

We now construct a lower solution of (3.13). Consider the boundary value problem

(3.16)

for .

By Theorem  3.1 in [12], there exists a solution for (3.16). We define an extension by for . Then, by Theorem  2.4 in [12] and Díaz-Saá's inequality in [24], we get

(3.17)

Setting and performing some standard computations, we see that ,

(3.18)

and in . Then, our result follows from Lemma 2.3.

We now give the proof of Theorem 2.4.

Proof of Theorem 2.4.

Let be a solution of the problem

(3.19)

where . We see that is an upper solution of the equation

(3.20)

if and only if

(3.21)

or

(3.22)

Since

(3.23)

we have a constant , such that

(3.24)

Denote

(3.25)

Since , we have , and there exist , such that for and for . Then . A simple computation shows that

(3.26)

Thus

(3.27)

Hence, for any , there exists a unique , such that . That is

(3.28)

Now defining , we get

(3.29)

This shows that is an upper solution of (3.20). Noting that

(3.30)

we know that is an upper solution of (1.1). Let be a solution of (3.11). Obviously, is a lower solution of (1.1). We now show that in .

Since for and as , then for any , there exist , such that . Without loss of generality, let .

From the proof of Lemma 3.4 and the definition of , we have for . Further, by (3.17), we get . Letting , we obtain in .

By Lemma 2.3, there exists a solution for the problem (1.1). We then complete the proof of Theorem 2.4.

Remark 3.5.

The nonlinear term can be regarded as a perturbation of the nonlinear term .

4. Proof of Nonexistence

In order to prove the nonexistence of nontrivial solution of the problem (2.21), we use the test function method, which has been used in [25] and references therein. Some modification has been made in our proof. The proof is based on argument by contradiction which involves a priori estimate for a nonnegative solution of (2.21) by carefully choosing the special test function and scaling argument.

Proof of Theorem 2.5.

Let be defined by

(4.1)

and put , by which the parameters will be determined later. It is not difficult to verify that and , where .

Suppose that is a solution to problem (2.21). Without loss of generality, we can assume that in (otherwise, we consider and let ). Let be a parameter ( will also be chosen below).

By the Young inequality, we get

(4.2)

where , , and satisfy (2.18) and , .

Multiplying the equation in (2.21) by and integrating by parts, we obtain

(4.3)

Then applying the Young inequality with parameter , we have

(4.4)

where .

Similarly, let us multiply the equation in (2.21) by and integrate by parts:

(4.5)

By (4.4),

(4.6)

Now, we apply the Hölder inequality to the integral on the right-hand side of (4.6):

(4.7)

with , and .

Since , we chose so small that . Then, we have

(4.8)

with , .

Since , with . Then we get

(4.9)

where and .

Let . Then,

(4.10)

Similarly,

(4.11)

Then it follows from (4.5)–(4.11) that

(4.12)

with and

(4.13)

If , it follows from (4.12) that

(4.14)

This implies that , a.e. in . That is, is a trivial solution for (2.21).

If , then (4.12) gives that

(4.15)

By (4.5), we derive

(4.16)

Reasoning as in the first part of the proof, we infer that

(4.17)

Letting in (4.17), we obtain (4.14). Thus, , a.e. in . Then the proof of Theorem 2.5 is completed.