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; linking
In this paper, we study the existence of multiple solutions to the gradient system
where is a bounded open domain with a smooth boundary ∂Ω and , λ is a real parameter and is fixed. is the set of all continuous, cooperative and symmetric matrix functions on . A matrix function takes the form
We impose the following assumptions on the function F:
Here and in the sequel, 0 is used to denote the origin in various spaces, and denote the norm and the inner product in , Bz denotes the matrix product in for a matrix B and . For two symmetric matrices B and C in , means that is positive definite.
and the associated norm
is well defined and is of class (see ) with derivatives
By () the system (GS)λ admits a trivial solution for any fixed parameter . We are interested in finding nontrivial solutions to (GS)λ. The existence of nontrivial solutions of (GS)λ depends on the behaviors of F near zero and infinity. The purpose of this paper is to find multiple nontrivial solutions to (GS)λ with superlinear term when the trivial solution acts as a local saddle point of the energy functional Φ in the sense that the parameter λ is close to a higher eigenvalue of the linear gradient system with the given weight matrix A
We will prove the following theorems.
We give some comments and comparisons. The superlinear problems have been studied extensively via variational methods since the pioneering work of Ambrosetti and Rabinowitz . 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 mountain-pass theorem in  and the cut-off techniques; and a third solution was constructed in a famous paper of Wang  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 mountain-pass theorem, built by Rabinowitz in  under a global sign condition (see ). Some extensions were done in [7,8]via local linking. More recently, in the work of Rabinowitz, Su and Wang , 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 . 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 is a saddle point of Φ.
We give some explanations regarding the conditions and conclusions. The assumptions ()-() are standard in the study of superlinear problems. () and () are used for bifurcation analysis. It sees that () implies that F is positive near zero, while () implies that F must be negative near zero. The local properties of F near zero are necessary for constructing homological linking. When , for any parameter λ in a bounded interval, say in , one can use the same arguments as in  to construct linking starting from . In our theorems, we do not require the global sign condition . When the parameter λ is close to the eigenvalue , the homological linking will be constructed starting from and this linking is different from the one in . This reveals the fact that when λ is close to from the right-hand side, the linking starting from can still be constructed even if F is negative somewhere. The conditions similar to () and () were first introduced in  where multiple periodic solutions for the second-order Hamiltonian systems were studied via the ideas in . Since we treat a different problem in the current paper, we need to present the detailed discussions although some arguments may be similar to those in [9,10].
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 1.1-1.3 are given in Section 5.
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 functional defined on a Hilbert space.
Let E be a Hilbert space and . Denote , , for . We say that Φ satisfies the (PS)c condition at the level if any sequence satisfying , as , has a convergent subsequence. Φ satisfies (PS) if Φ satisfies (PS)c at any .
is called the qth critical group of Φ at infinity (see ).
If , then for all q. Since for each , it follows that if for some , then Φ must have a critical point with . If , then for all q. Thus, if for some q, then Φ must have a new critical point. One can use critical groups to distinguish critical points obtained by other methods and use the Morse equality to find new critical points.
(2) for (Gromoll-Meyer );
We note here that under the framework of Proposition 2.3, and ∂Qhomotopically link with respect to the direct sum decomposition . and ∂Q are also homologically linked. The conclusion (2.3) follows from Theorems 1.1′ and 1.5 of Chapter II in . (See also .)
Then the following variational inequalities hold:
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.
is bounded in E. Here and below, we use C to denote various positive constants. We modify the arguments in . Choosing a positive number for n large, we have that
it follows that
we see that
The proof is complete. □
Now, we apply Proposition 2.3 to get the following existence result with partial homological information.
Since and ∂Q homotopically link with respect to the decomposition , and , it follows from Proposition 2.3 that Φ has a critical point with positive energy and its critical group satisfying (3.13). The proof is complete. □
We give some remarks. The existence of one nontrivial solution in Theorem 3.4 is valid when F is of class . From Lemma 3.2, one sees that the energy of the obtained solution is bounded away from 0 as λ is close to . A rough local sign condition on F is needed. If , then for any fixed , a linking with respect to can be constructed. Proposition 2.3 is applied again to get a nontrivial solution satisfying
Therefore, when a global sign condition is present, as λ is close to from the left-hand side, two linkings can be constructed and two nontrivial solutions can be obtained. The question is how to distinguish from . Theorem 3.4 includes the case that for λ close to from the right-hand side, the linking with respect to is constructed provided the negative values of F are small. This phenomenon was first observed in .
4 Solutions via bifurcation
Proposition 4.1 (Theorem 11.35 in )
Proof Under the assumptions ()-(), for each eigenvalue of (), is a bifurcation point of (GS)λ (see ).
Let () hold. By the elliptic regularity theory (see ), small implies small. Then by (), we have that
Now, consider the linear eigenvalue gradient system:
We denote the distinct eigenvalues of (4.6) by as . By (), if we take , then for each , there is such that . By (4.5), the standard variational characterization of the eigenvalues of (4.6) shows that is less than the corresponding jth ordered eigenvalue of (). Furthermore, as in E. By (4.3) and (4.4), we see that z is a solution of (4.6) with eigenvalue λ. It must be that since λ is close to . Therefore, the case (ii) of Proposition 4.1 occurs under the given conditions. This proves the case (1). The existence for the case (2) is proved in the same way.
Applying the elliptic regularity theory, we have that
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 )
The following arguments are from . As , it follows from (5.2) and (5.3) that for each , there is a unique such that
Now, Φ can be written as
Here we use a potential convention that (GS)λ has finitely many solutions and then 0 is isolated. Otherwise, one would have that as small, implies for all , for all . Thus, 0 would not be an isolated critical point of Φ and (GS)λ would have infinitely many nontrivial solutions. By (5.10) and (5.11), we verify that
Applying Proposition 2.2, we obtain
Finally, we prove the theorems.
From Proposition 2.1(2), we have that
By Lemma 5.1 and Lemma 5.2(3), we have that
Assume that (GS)λ has only two solutions 0 and . Choose such that . Then by the deformation and excision properties of singular homology (see ), we have
We deduce by (5.15) and (5.18) that
which contradicts (5.14). The proof is complete. □
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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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