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Multiplicity of solutions for singular semilinear elliptic equations in weighted Sobolev spaces
Boundary Value Problems volume 2014, Article number: 156 (2014)
Abstract
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.
1 Introduction
In this paper we investigate the following singular semilinear problem:
where
and is an open set (possibly unbounded).
In recent years, the existence and multiplicity of solutions for semilinear equations have been extensively studied by many authors (see [1]–[11]).
In 2001, Shapiro [12] 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 [13]–[15]). 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).
We introduce the pre-Hilbert space:
with the inner product
Let denote the Hilbert space completed by the norm , denote the Hilbert space with the inner product .
Next, we define the two-form
Definition 1.1
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.
Remark 1.1
where .
Definition 1.2
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 .
(): and
for .
(): For each (), , for , with the property that
where is in for , and also, to be quite explicit,
In fact, there are many examples which use special functions to illustrate the -region or the simple -region (see [12]).
The weak solutions of (1.1) correspond to the critical points of the functional defined by
where and .
We define
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.
(): There exists a function , , such that
(): There exists a constant such that
(): and .
(): There exists such that for the first eigenfunction
(): There exists such that for the first eigenfunction
(): There exist and such that
(): Let , a.e. , and
(): There exists such that
for some positive constant C.
():, for every , where denotes the eigenspace associated with the eigenvalue .
(): There exist and such that
Remark 1.2
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.
Theorem 1.1
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.
Theorem 1.2
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.
Theorem 1.3
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.
Theorem 1.4
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.
Theorem 1.5
Letbe a domain satisfying simple-region, and. Assume that. Furthermore, suppose the conditions () and (). Then the problem (1.1) has infinitely many solutions in.
Theorem 1.6
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.
Lemma 2.1
Let L be defined as (1.2). Then
for some positive constants, .
To get Lemma 2.7 and Lemma 2.8, we first introduce a corollary of the Ekeland variation principle (see [16], Theorem 2.4).
Lemma 2.2
Let X be a Banach space. Assume thatis bounded from below, which satisfies thecondition. Thenis a critical value.
Lemma 2.3
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, .
Proof
See Lemma 2 and Theorem 9 in [12]. □
To establish the multiplicity of solutions for problem (1.1), we need to apply the following fundamental theorem (see [17], Theorem 9.12).
Lemma 2.4
Let X be an infinite-dimensional Banach space and letbe continuous, even and satisfyingfor every. Assume, also, that:
-
(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.
Lemma 2.5
Under hypotheses () and (), is a-region, the functional I has the following saddle geometry for:
-
(1)
if with .
-
(2)
There is such that , .
-
(3)
, , where is given by ().
Proof
-
(1)
By Remark 1.1, we have
Using (), we have , as .
-
(2)
By a simple calculation, we get
By using () we have
So we choose .
-
(3)
By () and Remark 1.1, we get
the proof of this lemma is completed. □
Next, we will prove the Palais-Smale properties for the functional I. We recall that satisfies the Palais-Smale conditions at the level ( in short). For any sequence such that
as , the sequence possesses a convergent subsequence in E. Moreover, we say that I satisfies conditions when we have for all (see [16]).
Lemma 2.6
Assume the condition () holds, , andis a-region. Then the functional I has theconditions wheneveror.
Proof
We only prove the lemma for . For the case , we can use similar methods.
-
1.
The boundedness of sequence.
If this is not the case, there exists a unbounded sequence such that . Without loss of generality, we may assume that, as , the following expressions hold:
Define . Hence by Lemma 2.3, there is an with the following properties:
in ,
in ,
a.e. in Ω.
For any , it is obvious that . By the convergence of and (), we have
According to the definition of , we obtain . So, we suppose initially that . Because , it is obvious that , a.e. as .
Hence, taking , , , by Remark 1.1, we have
Since , it can easily be concluded that the sequence is bounded. Because of , on a subsequence , without loss of generality, we assume .
Now, by the Hölder inequality and (), we have
Thus, by applying the dominated convergence theorem, we conclude that
On the other hand,
by (2.2) and Remark 1.1, we obtain
as . Therefore, due to Remark 1.1, we have
Consequently, by virtue of Fatou’s lemma and assumption (), we get
which contradicts with the condition . Hence, the sequence of the functional I is bounded.
-
2.
Various convergences of .
Since is a bounded sequence, there is an with the following properties:
in ,
in ,
a.e. in Ω.
-
3.
converges to u in .
From the definition of sequence, we have, as ,
By Fatou’s lemma and the above convergence of , it is easy to show that
as . Hence, we get
By the weak convergence, it follows that
By using (2.3), (2.4), (2.5), and a simple calculation, we obtain
From Lemma 2.1, the proof is completed. □
Lemma 2.7
Suppose that () and () are satisfied, andis a simple-region. Then the origin is a local minimum for the functional I.
Proof
From (1.4), it is easy to see that . By (), we can choose and a constant such that
Consequently, by Remark 1.1 and Lemma 2.3, we have
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.
Lemma 2.8
Let the hypotheses (), (), (), and () hold, and letbe a-region. Then there existssuch thatand, where r is given by Lemma 2.7.
Proof
By () and (), we take where is provided by (). Thus, we obtain
and . By Lemma 2.7, we have , then the conclusion follows. □
Lemma 2.9
If hypotheses (), (), and () are satisfied, andis a-region. Then problem (1.1) has at least one nontrivial solution. Moreover, has negative energy, i.e. .
Proof
By () and Remark 1.1, we have
Therefore, the functional I is bounded below. In this case, we would like to mention that the functional I satisfies the conditions with . To see this, by Lemma 2.6, we take provided by (), we obtain
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.
Lemma 2.10
Assume that the conditions (), (), (), and () are satisfied, andis a-region. Then the problem (1.1) has at least two nontrivial solutions with negative energy.
Proof
Define
We have . Hence, we minimizer the functional I restrict to and .
Firstly, we consider the functionals . By the assumption (), we have
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 .
Moreover, we affirm that and are nonzero critical points. Based on () and (), we obtain
and I is restricted to being nonnegative. More specifically, given and (3) in Lemma 2.5, we have
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.
Lemma 2.11
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
It is easy to see that functional I is continuous and even. Moreover, by Lemma 2.11, I satisfies condition for every . To prove Theorem 1.6, we only need to test and verify the conditions (1) and (2) in Lemma 2.4.
-
1.
By () and Remark 1.1, we have
Hence there exist small enough and such that , for . That is to say the condition (1) in Lemma 2.4 holds with .
-
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)
By (), there exist satisfying a.e. and a positive constant c such that
The inequality (3.1) implies
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.
Lemma 4.1
Assume that the condition () holds, and thatis a-region. Then the functional I satisfies theconditions wheneveror.
Proof
We only prove the lemma for all . For the case , we can use similar methods.
The boundedness of sequence.
Let us prove by contradiction. Suppose that there exists a unbounded sequence such that . For ease of notation and without loss of generality, we assume that
Define . Hence by Lemma 2.3, there is an with the following properties:
For any , we have . By the convergence of and (), we have
By the definition of , we obtain . It is obvious that , as , and , as . By (), we have .
Hence, we can take , where , , . , denotes the eigenspace associated to the eigenvalue and denotes the eigenspace associated to the eigenvalue . Set
By Remark 1.1, we have
Hence, by Lemma 2.1, we get
and is bounded in , i.e., is bounded in . Letting , one has
By (), the Hölder inequality, and the Lebesgue dominated theorem, it is easy to show
Since and is bounded in , we have and
Consequently, by virtue of Fatou’s lemma, (4.1) and (), 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. □
Lemma 4.2
Suppose that () and () are satisfied, is a simple-region and. Then the origin is a local minimum for the functional I.
Proof
By (), we can choose and a constant such that
Consequently, by Remark 1.1 and Lemma 2.3, we have
, where r is small enough. Therefore the proof is completed. □
Proof of Theorem 1.6
Since
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. □
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Acknowledgements
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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Jia, G., Zhang, Lj. Multiplicity of solutions for singular semilinear elliptic equations in weighted Sobolev spaces. Bound Value Probl 2014, 156 (2014). https://doi.org/10.1186/s13661-014-0156-9
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DOI: https://doi.org/10.1186/s13661-014-0156-9