Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

Abstract

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

1. Introduction and Main Results

Consider the following semilinear elliptic equations with Dirichlet boundary value conditions: where is a smooth bounded domain in , is the Hardy-Sobolev critical exponent, is the Sobolev critical exponent, , and .

The energy functional associated with problem (1) is defined by for any . In general, a function is called a weak solution of problem (1) if and for all ; it holds

The paper by Crandall et al. [1] is the starting point on semilinear problem with singular nonlinearity. There is a large literature on singular nonlinearities (see [2–14] and references therein). In particular, the following Dirichlet problem has been shown in [2], in which the authors proved that problem (4) possesses at least one solution for small enough, and there exists no solution when is large. Chabrowski in [15] considered the Neumann boundary problems with singular superlinear nonlinearities by approximation and variational methods. When the superlinear term is subcritical, he obtained two solutions, a mountain-pass solution and a local minimizer solution. And, in the critical case, he obtained a local minimizer solution and proved that there is no moutain-pass solution.

In recent years, people have paid much attention to the existence of solutions for problems with the Sobolev critical exponent (the case that ) (see [16–21] and the references therein); some authors also considered the singular problems with the Hardy-Sobolev critical exponent (the case that ) (see [22–27] and the references therein).

Up to our knowledge, the literature does not contain any result on the existence of positive solutions to the problem (1) with the nonlinearity containing singularity and Hardy-Sobolev exponents. Motivated by reasons above, the aim of this paper is to show the existence of positive solutions of problem (1). We study problem (1) and obtain at least two solutions via the Nehari method. It is well-known that the singular term leads to the nondifferentiability of the functional on , so does not belong to . In order to get the existence of multiple positive solutions of problem (1), we use the Nehari method and differentiate the two solutions by their different Nehari-type sets.

The main result can be described as follows.

Theorem 1. Let and . Then, there exists small enough, such that problem has at least two positive solutions for any .

The paper is organized as follows: in Section 2, we give some preliminaries; in Section 3, we prove Theorem 1. This idea is essentially introduced in 20]. Throughout this paper, we make use of the following notations: (i)The norm in is denoted by

By Hardy inequality [28], we easily derive that the norm is equivalent to the usual norm: (ii) denotes the space of the functions such that endowed with scalar product and norm, respectively, given bythat coincides with the completion of with respect to the -norm of the gradient. By Hardy inequality [28], we easily derive that the norm is equivalent to the usual norm: in . (iii)The norm in is denoted by (iv) denote positive constants

2. Preliminaries

In this section, we will study the unperturbed problem

It is well-known that the nontrivial solutions of problem (9) are equivalent to the nonzero critical points of the energy functional

Obviously, the energy functional is well-defined and is of with derivatives given by

For all , it is well-known that the function solves equation (9) and satisfies

Moreover is the extremal function of the minimization problem

In view of [27, 29], we have the following exact local behavior of the solutions of (1).

Lemma 2. Let . If is a positive solution of (1), then there exists small enough and some positive constants and such thatDefine such that for any for all , and Denote . Then, using an argument similar to [30], we have the following lemma.

Lemma 3. Let be a weak solution of problem (1). Then, for small enoughNext, we define some Nehari-type sets, which are relevant in getting multiple positive solutions. Denote Define the minimization problems Since it is easy to see that for and as . Take such that for any . Denote and set

Lemma 4. If , then . Moreover, for any , there exists a unique such that andand there exists a unique such that and

Proof. The proof is similar to [30]. We omit the details.

3. Proof of Theorem 1

In this section, we will prove Theorem 1. The proof of Theorem 1 is based on solving the minimization problem (18) and the minimization problem

We divide the proof into two steps. In the first step, we prove that if there is such that and there is such that , then and are two positive weak solutions of (1). In the second step, we prove that the minima in (18) and in (23) are achieved, respectively.

Step 1. Let be such that and such that .

Lemma 5. For each and , we have the following:
There is such that for each
as , where for each is the unique positive number satisfying

Proof. The proof follows exactly the scheme in the proof of Lemma 3 in [31].

Lemma 6. For each and , we have and . Moreover,In particular, for all .

Proof. We only prove (24) since the proof of (25) is similar. Let and . By (i) of Lemma 5 and simple computations, we have that

Since the right hand side of the inequality has a finite limit value as , by direct calculations, we conclude increases monotonically as and

The Fatou lemma yields and we get (24).

Since and by the strong maximum principle, it follows that

Lemma 7. We have that and are positive weak solutions of (1).

Proof. For any and , we define and . Then, Since , we obtain from (24) that Dividing by and letting , by , we have the measure of which tends to 0 as , and we get that Therefore, Since is arbitrary, we get that is a solution of (1). Similarly, we can prove that is also a solution of (1).

Step 2. The minima and are achieved. The proof is exactly the same as [32]. We omit the details here.
We point out that and the exact local behavior of (see Lemma 2.) play essential roles. From Lemma 2., we have So there is such that for .

Lemma 8. Under the assumptions of Theorem 1,

Proof. First, using an argument similar to the proofs in [31], we have such that . It remains to be proven that

Since is a solution, we obtain by direct computation that

Note that the following inequality (see [31]) holds: there is and such that for each r≥m and t≥0.

By direct calculations, we can get that

So as and as . Hence, we only consider the right-hand side of the above inequality in the case of for some Hence, we obtain from Lemma 3 that

The proof is complete.

Lemma 9. The minimum in (23) is achieved by with .

Proof. Let be any sequence in such that . It is easy to see that is bounded in . Then, there exists a and a subsequence of , still denoted by such that Set and assume that Since , according to Br’zis-Lieb’s Lemma (see Lemma 1.32 in MW [33]) and the Sobolev embedding theorem, one gets We claim that . Indeed, if , then (since for any , is bounded away from zero) and this means that which contradicts the previous lemma.
From the assumption , by Lemma 4, we have such that and . For , we define Now, we consider the cases (i)(ii) and (iii) and Case (i). From , Define , then is increasing as ; is decreasing as so has maximum value As we have and , so we can see that is increasing on . Then, we have which is a contradiction.
Case (ii). We set . We know that attains the unique maximum at and . Moreover, for and for .
From the assumption on , we know . If , we have which contradicts the previous lemma. Thus, we have . By virtue of for , we obtain for and which also contradicts the previous lemma.
Case (iii). If , then we obtain from the fact that by some computations that and , from the definition of , which contradicts . So Hence, we have with .
The proof is complete.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11671331) and National Foundation Training Program of Jimei University (ZP2020057).