Abstract
In this paper, we study the multiplicity of nontrivial solutions for a superlinear gradient system with saddle structure at the origin. We make use of a combination of bifurcation theory, topological linking and Morse theory.
MSC: 35J10, 35J65, 58E05.
Keywords:
gradient system; superlinear; critical group; Morse theory; linking1 Introduction
In this paper, we study the existence of multiple solutions to the gradient system
where
with the functions
We impose the following assumptions on the function F:
(
(
(
(
for all
(
(
Here and in the sequel, 0 is used to denote the origin in various spaces,
Let E be the Hilbert space
and the associated norm
By the compact Sobolev embedding
is well defined and is of class
for
By (
It is known (see [2,3]) that for a given matrix
such that
Denote by
We will prove the following theorems.
Theorem 1.1Assume (
Theorem 1.2Assume (
Theorem 1.3Assume (
We give some comments and comparisons. The superlinear problems have been studied extensively via variational methods since the pioneering work of Ambrosetti and Rabinowitz [4]. Most known results on elliptic superlinear problems are contributed to a single equation with Dirichlet boundary data. Let us mention some historical progress on a single equation. When the trivial solution 0 acted as a local minimizer of the energy functional, one positive solution and one negative solution were obtained by using the mountainpass theorem in [4] and the cutoff techniques; and a third solution was constructed in a famous paper of Wang [5] by using a two dimensional linking method and a Morse theoretic approach. When the trivial solution 0 acted as a local saddle point of the energy functional, the existence of one nontrivial solution was obtained by applying a critical point theorem, which is now well known as the generalized mountainpass theorem, built by Rabinowitz in [6] under a global sign condition (see [1]). Some extensions were done in [7,8]via local linking. More recently, in the work of Rabinowitz, Su and Wang [9], multiple solutions have been obtained by combining bifurcation methods, Morse theory and homological linking when 0 is a saddle point in the sense that the parameter λ is very close to a higher eigenvalue of the related linear operator.
In the current paper, we build multiplicity results for superlinear gradient systems
by applying the ideas constructed in [9]. These results are new since, to the best of our knowledge, no multiplicity results
for gradient systems have appeared in the literature for the case that
We give some explanations regarding the conditions and conclusions. The assumptions
(
The paper is organized as follows. In Section 2, we collect some basic abstract tools. In Section 3, we get solutions by linking arguments and give partial estimates of homological information. In Section 4, we get solutions by bifurcation theorem and give the estimates of the Morse index. The final proofs of Theorems 1.11.3 are given in Section 5.
2 Preliminary
In this section, we give some preliminaries that will be used to prove the main results
of the paper. We first collect some basic results on the Morse theory for a
Let E be a Hilbert space and
We assume that Φ satisfies (PS) and
is called the qth critical group of Φ at
Let
is called the qth critical group of Φ at infinity (see [13]).
We call
If
For the critical groups of Φ at an isolated critical point, we have the following basic facts (see [11,12]).
Proposition 2.1Assume thatzis an isolated critical point of
(1)
(2)
(3) if
(4) if
Let 0 be an isolated critical point of
Then
The concept of local linking was introduced in [7]. In [15] a partial result was given for a
Now, we recall an abstract linking theorem which is from [1,12,15].
LetEbe a real Banach space with
(
where
(
where
Then Φ has a critical point
We note here that under the framework of Proposition 2.3,
We finally collect some properties of the eigenvalue problem (
The compactness of
(
and each eigenvalue
Set
Then the following variational inequalities hold:
We refer to [2,3] for more properties related to the eigenvalue problem (
3 Solutions via homological linking
In this section, we give the existence a nontrivial solution of (GS)_{λ} by applying homological linking arguments and then give some estimate of its Morse index. The following lemmas are needed.
Lemma 3.1Assume thatFsatisfies (
Proof By (
is bounded in E. Here and below, we use C to denote various positive constants. We modify the arguments in [1]. Choosing a positive number
By (
Therefore,
where
Thus, for a fixed
Therefore,
Now, we construct a homological linking with respect to the direct sum decomposition
of E for
Take an eigenvector
Lemma 3.2Assume thatFsatisfies (
Proof By the conditions (
where
it follows that
where
Since
on
we see that
Since
The constants α and ρ are independent of
Lemma 3.3Assume thatFsatisfies (
where
Proof From (
Since
Now, fix such an
Since
then, when
The proof is complete. □
Now, we apply Proposition 2.3 to get the following existence result with partial homological information.
Theorem 3.4LetFsatisfy (
Proof By Lemma 3.1, Φ satisfies (PS). By Lemmas 3.2 and 3.3, for each fixed
Since
We give some remarks. The existence of one nontrivial solution in Theorem 3.4 is valid
when F is of class
Therefore, when a global sign condition
4 Solutions via bifurcation
In this section, we get two solutions for (GS)_{λ}via bifurcation arguments [1]. We first cite the bifurcation theorem in [1].
Proposition 4.1 (Theorem 11.35 in [1])
LetEbe a Hilbert space and
where
Let
(i)
(ii) there is a onesided neighborhood Λ ofμsuch that for all
(iii) there is a neighborhood Λ ofμsuch that for all
We apply Proposition 4.1 to get two nontrivial solutions of (GS)_{λ} for λ close to an eigenvalue of (
Theorem 4.2Assume thatFsatisfies (
(1) every
(2) every
Furthermore, the Morse index
Proof Under the assumptions (
Let
By (
Let (
Now, consider the linear eigenvalue gradient system:
We denote the distinct eigenvalues of (4.6) by
Now, we estimate the Morse indices for the solutions obtained above. Let
Applying the elliptic regularity theory, we have that
For each
Therefore, for
and for
By (
Therefore, the Morse index
5 Proofs of main theorems
In this section, we give the proof of main theorems in this paper. We first compute the critical groups of Φ at both infinity and zero.
Lemma 5.1 (see [5])
LetFsatisfy (
Proof The idea of the proof comes from the famous paper [5]. We include a sketched proof in an abstract version. Given
The following arguments are from [10]. As
By (5.4) and the implicit function theorem, we have that
Then
Clearly, ϱ is continuous, and for all
Therefore,
and so
since
Lemma 5.2LetFsatisfy (
(1) For
(2) For
(3) For
(4) For
Proof By (
(1) When
(2) When
(3) When
Assume that
Now, Φ can be written as
By (
Hence, for
Since
By (
For
For
For
Since
For
Here we use a potential convention that (GS)_{λ} has finitely many solutions and then 0 is isolated. Otherwise, one would have that
as
Applying Proposition 2.2, we obtain
(4) When
Finally, we prove the theorems.
Proof of Theorem 1.1 It follows from (
By Theorem 4.2(1), (GS)_{λ} has two nontrivial solutions
From Proposition 2.1(2), we have that
From (5.12) and (5.13), we see that
Proof of Theorem 1.2 With the same argument as above, it follows from Theorem 4.2(2) and Theorem 3.4 for
the part
Proof of Theorem 1.3 By Theorem 3.4 for the part
By Lemma 5.1 and Lemma 5.2(3), we have that
Assume that (GS)_{λ} has only two solutions 0 and
By (5.17), the long exact sequences for the topological triple
We deduce by (5.15) and (5.18) that
Take
which contradicts (5.14). The proof is complete. □
We finally remark that Theorem 1.1 is valid for
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the anonymous referee for his/her valuable suggestions. The second author was supported by NSFC11271264, NSFC11171204, KZ201010028027 and PHR201106118.
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