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
with the functions satisfying the conditions that for all , which means A is cooperative and that .
We impose the following assumptions on the function F:
( ) .
( ) , , for .
( ) There is and such that
( ) There is , such that
for all , .
( ) for small and .
( ) for small and .
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.
Let E be the Hilbert space endowed with the inner product
and the associated norm
By the compact Sobolev embedding for , under the assumptions ( ) and ( ), the functional
is well defined and is of class (see ) with derivatives
for , , , . Therefore, the solutions to (GS)λ are exactly critical points of Φ in E.
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
such that as .
Denote by the negative part of F, i.e., .
We will prove the following theorems.
Theorem 1.1Assume ( )-( ), ( ) and let be fixed. Then there is such that when , for all , (GS)λ has at least three nontrivial solutions inE.
Theorem 1.2Assume ( )-( ), ( ) and let be fixed. Then there is such that when , for all , (GS)λ has at least three nontrivial solutions inE.
Theorem 1.3Assume ( )-( ) and for , with small. Then there is such that when , for all , (GS)λ has at least two nontrivial solutions inE.
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 .
We assume that Φ satisfies (PS) and . Let with and U be a neighborhood of such that . The group
Let . The group
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.
Proposition 2.1Assume thatzis an isolated critical point of with a finite Morse index and nullity . Then
(1) if ;
(2) for (Gromoll-Meyer );
(3) if then ;
(4) if then .
Let 0 be an isolated critical point of with a finite Morse index and nullity . Assume that Φ has a local linking at 0 with respect to a direct sum decomposition , , i.e., there exists small such that
Then for either or .
LetEbe a real Banach space with and be finite. Suppose that satisfies (PS) and
( ) there exist and such that
( ) there exist and with such that
Then Φ has a critical point with and
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 .)
We finally collect some properties of the eigenvalue problem ( ). Associated with a matrix , there is a compact self-adjoint operator such that
The compactness of follows from the compact embedding . The operator possesses the property that is an eigenvalue of ( ) if and only if there is nonzero such that
( ) has the sequence of distinct eigenvalues
and each eigenvalue of ( ) has a finite multiplicity. For , denote
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.
Lemma 3.1Assume thatFsatisfies ( )-( ), then for any fixed , the functional Φ satisfies the (PS) condition.
Proof By ( ) and the compact embedding for , it is enough to show that any sequence with
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
By ( ) we deduce that
where . By the Hölder inequality and the Young inequality, we get for any that
Thus, for a fixed small enough, we have by (3.3) that
Therefore, is bounded in E. The proof is complete. □
Now, we construct a homological linking with respect to the direct sum decomposition of E for :
Take an eigenvector corresponding to the eigenvalue of ( ) with . Set
Lemma 3.2Assume thatFsatisfies ( )-( ) and . Then there exist constants small, such that for all , such that
Proof By the conditions ( ) and ( ), for , there is such that
where is the constant for the embedding such that for . Since for ,
it follows that
where is independent of λ and
Since and the function achieves its maximum
we see that
Since is a decreasing function with respect to λ for any fixed small, (3.4) holds for
The constants α and ρ are independent of . The proof is complete. □
Lemma 3.3Assume thatFsatisfies ( ), ( ) and . Then there exist , and , all independent ofλ, such that when and ,
Proof From ( ) we deduce (3.2) with a positive constant C independent of λ. For , write , , , . Assume that , then
Since and , (3.10) shows that there exists independent of λ such that
Now, fix such an that with ρ given in Lemma 3.2. For with , we write , , . Set . Then we have that
Since , taking
then, when and ,
The proof is complete. □
Now, we apply Proposition 2.3 to get the following existence result with partial homological information.
Theorem 3.4LetFsatisfy ( )-( ) and . Then there is such that when , for each , (GS)λ has one nontrivial solution with a critical group satisfying
Proof By Lemma 3.1, Φ satisfies (PS). By Lemmas 3.2 and 3.3, for each fixed , Φ satisfies ( ) and ( ) in the sense that
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 )
LetEbe a Hilbert space and with
where is symmetric and as . Consider the equation
Let be an isolated eigenvalue of finite multiplicity. Then either
(i) is not an isolated solution of (4.1) in , or
(ii) there is a one-sided neighborhood Λ ofμsuch that for all , (4.1) has at least two distinct nontrivial solutions, or
(iii) there is a neighborhood Λ ofμsuch that for all , (4.1) has at least one nontrivial solution.
We apply Proposition 4.1 to get two nontrivial solutions of (GS)λ for λ close to an eigenvalue of ( ) and then give the estimates of the Morse index.
Theorem 4.2Assume thatFsatisfies ( )-( ). Let be fixed. Then there exists such that (GS)λ has at least two nontrivial solutions for
(1) every if ( ) holds;
(2) every if ( ) holds.
Furthermore, the Morse index and the nullity of such a solution satisfy
Proof Under the assumptions ( )-( ), for each eigenvalue of ( ), is a bifurcation point of (GS)λ (see ).
Let be a solution of (GS)λ near which satisfies
By ( ) and ( ), we have
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.
Now, we estimate the Morse indices for the solutions obtained above. Let be a bifurcation solution of (GS)λ. Then
Applying the elliptic regularity theory, we have that
For each , we have
Therefore, for ,
and for ,
By ( ) and (4.7), there exists such that when ,
Therefore, the Morse index and the nullity of satisfy (4.2). The proof is complete. □
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 )
LetFsatisfy ( )-( ), then for any fixed ,
The following arguments are from . As , it follows from (5.2) and (5.3) that for each , there is a unique such that
By (5.4) and the implicit function theorem, we have that . Define
Then . Define a map by
Clearly, ϱ is continuous, and for all with , by (5.4),
and so is a strong deformation retract of . Hence,
since is contractible, which follows from the fact that . □
Lemma 5.2LetFsatisfy ( )-( ).
(1) For , .
(2) For , .
(3) For , if for small, then .
(4) For , if for small, then .
Proof By ( ), we have
(1) When , is a nondegenerate critical point of Φ with the Morse index , thus .
(2) When , is a nondegenerate critical point of Φ with the Morse index , thus .
(3) When , is a degenerate critical point of Φ with the Morse index and the nullity , .
Assume that for with small. We will show that Φ has a local linking structure at with respect to . If this has been done, then by Proposition 2.2, we have .
Now, Φ can be written as
By ( ) and ( ), for , there is such that
Hence, for , we have that
Since is finite dimensional, all norms on are equivalent, hence for small,
By ( ), we have that for some ,
For , we write where and . Then
For with , we have . Hence, by (5.6) and the Poincaré inequality, we have for various constants ,
For with , . Therefore,
Since , for small,
For , it must hold that
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
(4) When for small, a similar argument shows that Φ has a local linking structure at with respect to . By Proposition 2.2, it follows that . □
Finally, we prove the theorems.
Proof of Theorem 1.1 It follows from ( ) that for small. By Theorem 3.4 for the part , (GS)λ has a nontrivial solution satisfying
By Theorem 4.2(1), (GS)λ has two nontrivial solutions ( ) with their Morse indices satisfying
From Proposition 2.1(2), we have that
From (5.12) and (5.13), we see that ( ). The proof is complete. □
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 . We omit the details. □
Proof of Theorem 1.3 By Theorem 3.4 for the part , (GS)λ has a solution with its energy and
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
By (5.17), the long exact sequences for the topological triple read as
We deduce by (5.15) and (5.18) that
Take in (5.19), then
which contradicts (5.14). The proof is complete. □
We finally remark that Theorem 1.1 is valid for , from which one sees that is a local minimizer of Φ.
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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