Multiplicity of solutions for singular semilinear elliptic equations in weighted Sobolev spaces
© Jia and Zhang; licensee Springer 2014
Received: 18 January 2014
Accepted: 7 June 2014
Published: 24 September 2014
A class of semilinear elliptic equations involving strong resonance or non-resonance is reconsidered here. The multiplicity of solutions is investigated by using the variational method, and the results complement earlier ones.
MSC: 35B34, 35J20, 35J61, 46E35.
and is an open set (possibly unbounded).
In 2001, Shapiro  studied a series of eigenvalue problems of singular quasilinear elliptic and parabolic equations in weighted Sobolev spaces, he obtained many existence results by using Galerkin method. Then Jia et al. continued to study the quasilinear elliptic equations (see –). However, their main results are only concerned with the existence of solutions without considering multiplicity of solutions.
Motivated by previous work, the aim of this paper is to establish the existence of infinitely many solutions of the elliptic equations in weighted Sobolev spaces, by using the Ekeland variation principle and the mountain-pass lemma.
We first assume that
():, , .
():, , , and there exists such that , .
():, is a positive constant.
Let designate a fixed closed set (Γ may be empty).
Let denote the Hilbert space completed by the norm , denote the Hilbert space with the inner product .
We say is a -region if the following two facts obtain:
(): There exists a complete orthonormal system in , and . Also .
(): There exists a sequence of eigenvalues with such that , . Also in Ω.
We say is a simple -region if is a -region and the following four conditions are satisfied:
(): Associated with each there are positive functions and in satisfying , for .
In fact, there are many examples which use special functions to illustrate the -region or the simple -region (see ).
where and .
Here, the above functions belong to and the limits are taken a.e. and uniformly in .
As for and , we make the assumptions as follows.
(): and .
for some positive constant C.
():, for every , where denotes the eigenspace associated with the eigenvalue .
We must find the function satisfying ()-(). Let , , and be the first eigenvalue and second eigenvalue of Laplacian eigenvalue problem of homogeneous boundary condition on Ω, and the first eigenfunction be . Let . For , , by simple calculation, if is large enough, we find that satisfies ()-().
On the other hand, if we take , where , then satisfies () and ().
The main results of our paper are given by the following theorems.
Letbe a domain satisfying the requirements of being a-region. Letin (1.1). Assume that. Furthermore, if the conditions (), (), and () are satisfied, then the problem (1.1) has at least one nontrivial solution in.
Letbe a domain satisfying-region. Andin (1.1). Assume that. Furthermore, if the conditions ()-() and () are satisfied, then the problem (1.1) has at least two nontrivial solutions in.
Letbe a domain satisfying simple-region, , and. Assume that. Furthermore, if the conditions (), (), (), and () are satisfied, then the problem (1.1) has at least two nontrivial solutions in.
Letbe a domain satisfying simple-region, , and. Assume that. Furthermore, if the conditions ()-(), (), and () are satisfied, then the problem (1.1) has at least three nontrivial solutions in.
Letbe a domain satisfying simple-region, and. Assume that. Furthermore, suppose the conditions () and (). Then the problem (1.1) has infinitely many solutions in.
Letbe a domain satisfying simple-region, . (). Assume that. Furthermore, suppose the conditions (), (), and (). Then the problem (1.1) has two nontrivial solutions in.
The paper is organized as follows. In Section 2, we provide and establish some lemmas which are necessary in the proof of our main theorems. In Section 3, we will prove Theorem 1.1-Theorem 1.5. In the last section, we establish Lemma 4.1 and Lemma 4.2, and we give the proof of Theorem 1.6.
2 Preliminary results
In this section, we prove some lemmas which will be used in the proof of our main theorems. For simplicity, we denote by in the following.
for some positive constants, .
To get Lemma 2.7 and Lemma 2.8, we first introduce a corollary of the Ekeland variation principle (see , Theorem 2.4).
Let X be a Banach space. Assume thatis bounded from below, which satisfies thecondition. Thenis a critical value.
Assume that L is given by (1.2), and the assumptions ()-() hold, and thatis a-region. Thenis compactly embedding in. Moreover, ifis a simple-region and, then for, is compactly embedded in, .
See Lemma 2 and Theorem 9 in . □
To establish the multiplicity of solutions for problem (1.1), we need to apply the following fundamental theorem (see , Theorem 9.12).
- (1)There exist , , and a subspace of finite codimension such that
- (2)For every finite-dimensional subspace , , there exists such that
Then there exists a sequenceof critical values of E with.
Next, we describe some results under the geometry for functional I.
if with .
There is such that , .
, , where is given by ().
- (1)By Remark 1.1, we have
- (2)By a simple calculation, we get
- (3)By () and Remark 1.1, we get
the proof of this lemma is completed. □
as , the sequence possesses a convergent subsequence in E. Moreover, we say that I satisfies conditions when we have for all (see ).
Assume the condition () holds, , andis a-region. Then the functional I has theconditions wheneveror.
The boundedness of sequence.
Define . Hence by Lemma 2.3, there is an with the following properties:
a.e. in Ω.
According to the definition of , we obtain . So, we suppose initially that . Because , it is obvious that , a.e. as .
Since , it can easily be concluded that the sequence is bounded. Because of , on a subsequence , without loss of generality, we assume .
Various convergences of .
Since is a bounded sequence, there is an with the following properties:
a.e. in Ω.
converges to u in .
From Lemma 2.1, the proof is completed. □
Suppose that () and () are satisfied, andis a simple-region. Then the origin is a local minimum for the functional I.
where r is small enough and , is provided by (). Therefore the proof is completed. □
To complete the mountain-pass geometry, we also need the following result.
Let the hypotheses (), (), (), and () hold, and letbe a-region. Then there existssuch thatand, where r is given by Lemma 2.7.
and . By Lemma 2.7, we have , then the conclusion follows. □
If hypotheses (), (), and () are satisfied, andis a-region. Then problem (1.1) has at least one nontrivial solution. Moreover, has negative energy, i.e. .
Consequently, applying Lemma 2.2 we have one critical point such that . The proof of this lemma is completed. □
To prove Theorem 1.3, we establish the following lemma.
Assume that the conditions (), (), (), and () are satisfied, andis a-region. Then the problem (1.1) has at least two nontrivial solutions with negative energy.
We have . Hence, we minimizer the functional I restrict to and .
By Lemma 2.6, satisfy the conditions whenever . Therefore, we find that satisfy the conditions with .
In this way, by using Lemma 2.2 for the functionals and , we obtain two critical points denoted by and , respectively. Thus, we have and .
Next, we prove that and are distinct. The proof of this affirmation is by contradiction. If , then . By (2.6), we obtain . Therefore, we have a contradiction. Consequently, we get . Thus the problem (1.1) has at least two nontrivial solutions. Moreover, these solutions have negative energy. □
To prove Theorem 1.5, we need the following lemma. The proof is similar to that of Lemma 2.6.
Assume that the conditions () and () hold, , andis a simple-region. Then functional I satisfies theconditions.
3 Proofs of Theorem 1.1-Theorem 1.5
In this section, we prove Theorems 1.1, 1.2, 1.3, 1.4, and 1.5.
Proof of Theorem 1.1
By Lemma 2.9, we get a solution which satisfies . It follows that the problem (1.1) has at least one nontrivial solution. The proof is completed. □
Proof of Theorem 1.2
Using Lemma 2.10, we obtain two distinct critical points and such that and . Therefore, the problem (1.1) has at least two nontrivial solutions. The proof is completed. □
Proof of Theorem 1.3
From Lemma 2.7 and Lemma 2.8, we know that the functional I satisfies the geometric conditions of the mountain-pass theorem. Moreover, the functional I satisfies the conditions for all . Thus, we have a solution given by the mountain-pass theorem. Obviously, the solution satisfies .
On the other hand, by Lemma 2.9, we get another solution satisfying . It follows that the problem (1.1) has at least two nontrivial solutions. □
Proof of Theorem 1.4
The conditions (), (), (), and () imply that Lemma 2.7 and Lemma 2.8 hold. Thus, we have one solution which satisfies .
On the other hand, using Lemma 2.10, we obtain two distinct critical points such that . In other words, the problem (1.1) has at least three nontrivial solutions. The proof is completed. □
Proof of Theorem 1.5
- 1.By () and Remark 1.1, we have
- 2.We verify the condition (2) in Lemma 2.4. Let W be a finite-dimensional subspace of . Let such that , i.e.(3.1)
Since is a norm on W and W is finite-dimensional, then, by (3.2), there exists such that . Since , we deduce that the set is bounded in and the condition (2) in Lemma 2.4 holds. □
4 Proof of Theorem 1.6
In this section, we consider the problem (1.1) in . In order to prove Theorem 1.6, we first establish the following lemmas.
Assume that the condition () holds, and thatis a-region. Then the functional I satisfies theconditions wheneveror.
We only prove the lemma for all . For the case , we can use similar methods.
The boundedness of sequence.
By the definition of , we obtain . It is obvious that , as , and , as . By (), we have .
which contradicts with the condition . Hence, the sequence of the functional I is bounded.
We can argue as the proof of Lemma 2.6, so the proof has been completed. □
Suppose that () and () are satisfied, is a simple-region and. Then the origin is a local minimum for the functional I.
, where r is small enough. Therefore the proof is completed. □
Proof of Theorem 1.6
as , we can choose and , satisfying and such that and , where r is given by Lemma 4.2.
On the other hand, since , we know and . Let and . Consider the functionals , which are the restrictions of I on . By Lemma 4.1 and (), for , we get satisfying conditions. So by the mountain-pass lemma, we get two critical points . It is obvious that . The proof has been completed. □
The authors express their sincere thanks to the referees for their valuable criticisms of the manuscript and for helpful suggestions. This work was supported by National Natural Science Foundation of China (11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).
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