Superlinear gradient system with a parameter

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.


Introduction
In this paper, we study the existence of multiple solutions to the gradient system with the functions a, b, c ∈ C(¯ , R) satisfying the conditions that b(x) ≥  for all x ∈¯ , which means A is cooperative and that max x∈¯ {a, c} > . We impose the following assumptions on the function F: (F  ) There is C >  and  < p < N N- :=  * such that (F  ) There is μ > , M >  such that for all x ∈ , |z| ≥ M. (F  ) F z (x, z) >  for |z| >  small and x ∈ . http://www.boundaryvalueproblems.com/content/2012/1/110 (F  ) F z (x, z) <  for |z| >  small and x ∈ .
Here and in the sequel,  is used to denote the origin in various spaces, | · | and (·, ·) denote the norm and the inner product in R  , Bz denotes the matrix product in R  for a  ×  matrix B and z = (u, v) ∈ R  . For two symmetric matrices B and C in R  , B > C means that B -C is positive definite.
Let E be the Hilbert space H   ( ) × H   ( ) endowed with the inner product z, w = (u, v), (φ, ψ) = (∇u∇φ + ∇v∇ψ) dx, z = (u, v), w = (φ, ψ) ∈ E and the associated norm By the compact Sobolev embedding E → L p ( ) × L p ( ) for p ∈ [,  * ), under the assumptions (F  ) and (F  ), the functional is well defined and is of class C  (see []) with derivatives (z), w = ∇z∇wλ A(x)z, w dx -∇F(x, z), w dx, for z = (u, v), w = (φ, ψ), z  = (u  , v  ), z  = (u  , v  ) ∈ E. Therefore, the solutions to (GS) λ are exactly critical points of in E. By (F  ) the system (GS) λ admits a trivial solution z =  for any fixed parameter λ ∈ R. 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 z =  acts as a local saddle point of the energy functional in the sense that the parameter λ is close to a higher eigenvalue of the linear gradient system with the given It is known (see [, ]) that for a given matrix A ∈ M  ( ), (L A ) admits a sequence of distinct eigenvalues of finite multiplicity such that λ A k → ∞ as k → ∞. http://www.boundaryvalueproblems.com/content/2012/1/110 Denote by Fthe negative part of F, i.e., F -(x, z) = max{-F(x, z), }. We will prove the following theorems.
Theorem . Assume (F  )-(F  ), (F  ) and let k ≥  be fixed. Then there is δ >  such that 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  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  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 [, ] 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  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 z =  is a saddle point of .
We give some explanations regarding the conditions and conclusions. The assumptions (F  )-(F  ) are standard in the study of superlinear problems. (F  ) and (F  ) are used for bifurcation analysis. It sees that (F  ) implies that F is positive near zero, while (F  ) implies that F must be negative near zero. The local properties of F near zero are necessary for constructing homological linking. When F ≥ , for any parameter λ in a bounded interval, say in [λ A k , λ A k+ ), one can use the same arguments as in [] to construct linking starting from λ A k+ . In our theorems, we do not require the global sign condition F ≥ . When the parameter λ is close to the eigenvalue λ A k+ , the homological linking will be constructed starting from λ A k+ and this linking is different from the one in []. This reveals the fact that when λ is close to λ A k+ from the right-hand side, the linking starting from λ A k+ can still be constructed even if F is negative somewhere. The conditions similar to (F  ) and (F  ) were first introduced in [] where multiple periodic solutions for the second-order http://www.boundaryvalueproblems.com/content/2012/1/110 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 [, ].
The paper is organized as follows. In Section , we collect some basic abstract tools. In Section , we get solutions by linking arguments and give partial estimates of homological information. In Section , we get solutions by bifurcation theorem and give the estimates of the Morse index. The final proofs of Theorems .-. are given in Section .

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 C  functional defined on a Hilbert space.
Let E be a Hilbert space and ∈ C  (E, R).
We assume that satisfies (PS) and is called the qth critical group of at z  , where H * (A, B) denotes a singular relative homology group of the pair (A, B) with coefficients field F (see [, ]). Let a < inf (K). The group is called the qth critical group of at infinity (see []). We call M q := z∈K dim C q ( , z) the qth Morse-type numbers of the pair (E, a ) and β q := dim C q ( , ∞) the Betti numbers of the pair (E, a ). The core of the Morse theory [, ] is the following relations between M q and β q : Thus, if C q ( , ∞) ∼ = C q ( , z * ) for some q, then 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 . Assume that z is an isolated critical point of ∈ C  (E, R) with a finite Morse index m(z) and nullity n(z). Then
Morse index m  and nullity n  . Assume that has a local linking at  with respect to a direct sum decomposition Then has a critical point z * with (z * ) = c * ≥ α and We note here that under the framework of Proposition ., S ρ and ∂Q homotopically link with respect to the direct sum decomposition E = X ⊕ Y . S ρ and ∂Q are also homologically linked. The conclusion (.) follows from Theorems . and . of Chapter II in []. (See also []. ) We finally collect some properties of the eigenvalue problem (L A ). Associated with a matrix A ∈ M  ( ), there is a compact self-adjoint operator T A : E → E such that The compactness of T A follows from the compact embedding E → L  ( ) × L  ( ). The operator T A possesses the property that λ A is an eigenvalue of (L A ) if and only if there is nonzero z ∈ E such that (L A ) has the sequence of distinct eigenvalues and each eigenvalue λ A of (L A ) has a finite multiplicity. For j ∈ N, denote Then the following variational inequalities hold: We refer to [, ] for more properties related to the eigenvalue problem (L A ) and the operator T A .

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 . Assume that F satisfies (F  )-(F  ), then for any fixed λ ∈ R, the functional satisfies the (PS) condition.
Proof By (F  ) and the compact embedding E → L p ( ) × L p ( ) for  ≤ p <  * , it is enough to show that any sequence {z n } ⊂ E 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 β ∈ (/μ, /) for n large, we have that C + β z n ≥ (z n )β (z n ), z n . http://www.boundaryvalueproblems.com/content/2012/1/110 By (F  ) we deduce that By the Hölder inequality and the Young inequality, we get for any >  that Thus, for a fixed >  small enough, we have by (.) that Therefore, {z n } is bounded in E. The proof is complete. Now, we construct a homological linking with respect to the direct sum decomposition of E for k ≥ : Take an eigenvector φ k+ corresponding to the eigenvalue λ A k+ of (L A ) with φ k+ = . Set Proof By the conditions (F  ) and (F  ), for > , there is C >  such that where S  is the constant for the embedding E → L  ( ) × L  ( ) such that |z|   ≤ S  z  for z ∈ E. Since for z ∈ E ⊥ k+ , it follows that where C is independent of λ and Since p >  and the function g(r) =   η(λ, )r  -C r p achieves its maximum Since η(λ, ) is a decreasing function with respect to λ for any fixed >  small, (.) holds for The constants α and ρ are independent of λ ≤ λ * . The proof is complete.
We give some remarks. The existence of one nontrivial solution in Theorem . is valid when F is of class C  . From Lemma ., one sees that the energy of the obtained solution is bounded away from  as λ is close to λ A k+ . A rough local sign condition on F is needed. If F ≥ , then for any fixed λ ∈ [λ A k , λ A k+ ), a linking with respect to E k ⊕ E ⊥ k can be constructed. Proposition . is applied again to get a nontrivial solution z * satisfying Therefore, when a global sign condition F ≥  is present, as λ is close to λ A k+ from the lefthand side, two linkings can be constructed and two nontrivial solutions can be obtained. The question is how to distinguish z * from z * . Theorem . includes the case that for λ close to λ A k+ from the right-hand side, the linking with respect to E k+ ⊕ E ⊥ k+ is constructed provided the negative values of F are small. This phenomenon was first observed in [].

Solutions via bifurcation
In this section, we get two solutions for (GS) λ via bifurcation arguments []. We first cite the bifurcation theorem in []. We apply Proposition . to get two nontrivial solutions of (GS) λ for λ close to an eigenvalue of (L A ) and then give the estimates of the Morse index. http://www.boundaryvalueproblems.com/content/2012/1/110 Theorem . Assume that F satisfies (F  )-(F  ). Let k ≥  be fixed. Then there exists δ >  such that (GS) λ has at least two nontrivial solutions for holds. Furthermore, the Morse index m(z λ ) and the nullity n(z λ ) of such a solution z λ satisfy Proof Under the assumptions (F  )-(F  ), for each eigenvalue λ A j of (L A ), (λ A j , ) is a bifurcation point of (GS) λ (see []).
(.) By (F  ) and (F  ), we have Let (F  ) hold. By the elliptic regularity theory (see []), z >  small implies z C >  small. Then by (F  ), we have that Now, consider the linear eigenvalue gradient system: We denote the distinct eigenvalues of (.) by μ  (z) < μ  (z) < · · · < μ i (z) < · · · as z = . By (F  ), if we take z = , then for each i ∈ N, there is j ∈ N such that μ i () = λ A j . By (.), the standard variational characterization of the eigenvalues of (.) shows that μ i (z) is less than the corresponding jth ordered eigenvalue λ A j of (L A ). Furthermore, μ i (z) → λ A j as z →  in E. By (.) and (.), we see that z is a solution of (.) with eigenvalue λ. It must be that λ < λ A k+ since λ is close to λ A k+ . Therefore, the case (ii) of Proposition . occurs under the given conditions. This proves the case (). The existence for the case () is proved in the same way. Now, we estimate the Morse indices for the solutions obtained above. Let z λ be a bifurcation solution of (GS) λ . Then Applying the elliptic regularity theory, we have that (  .  ) http://www.boundaryvalueproblems.com/content/2012/1/110 For each y ∈ E, we have (z λ )y, y = Q λ (y) -F z (x, z λ )y, y dx.

Proofs of main theorems
In this section, we give the proof of main theorems in this paper. We first compute the critical groups of at both infinity and zero.
The following arguments are from []. As () = , it follows from (.) and (.) that for each z ∈ S  , there is a unique τ (z) >  such that By (.) and the implicit function theorem, we have that τ ∈ C(S  , R). Define Then π ∈ C(E \ {}, R). Define a map : Clearly, is continuous, and for all z ∈ E \ {} with (z) > a, by (.), Therefore, and so a is a strong deformation retract of E \ {}. Hence, since S  is contractible, which follows from the fact that dim E = ∞.
() When λ ∈ (λ A k , λ A k+ ), z =  is a nondegenerate critical point of with the Morse index m  = k , thus C q ( , ) ∼ = δ q, k F.
() When λ = λ A k+ , z =  is a degenerate critical point of with the Morse index m  = k and the nullity n  = dim E(λ A k+ ), m  + n  = k+ . Assume that F(x, z) ≤  for |z| ≤ σ with σ >  small. We will show that has a local linking structure at z =  with respect to E = E k ⊕ E ⊥ k . If this has been done, then by Proposition ., we have C q ( , ) ∼ = δ q, k F. Now, can be written as By (F  ) and (F  ), for > , there is C >  such that Hence, for z ∈ E k , we have that Since E k is finite dimensional, all norms on E k are equivalent, hence for r >  small, By (F  ), we have that for some C > , Hence, by (.) and the Poincaré inequality, we have for various constants C > , Since p > , for r >  small, For z = y ∈ E(λ A k+ ), it must hold that Here we use a potential convention that (GS) λ has finitely many solutions and then  is isolated. Otherwise, one would have that as r >  small, y ≤ r implies |y(x)| ≤ δ for all http://www.boundaryvalueproblems.com/content/2012/1/110 x ∈ , ∇F(x, y) ≡  for all x ∈ . Thus,  would not be an isolated critical point of and (GS) λ would have infinitely many nontrivial solutions. By (.) and (.), we verify that (z) > , for z ∈ E ⊥ k ,  < z ≤ r.
() When F(x, z) ≥  for |z| small, a similar argument shows that has a local linking structure at z =  with respect to E = E k+ ⊕ E ⊥ k+ . By Proposition ., it follows that C q ( , ) ∼ = δ q, k+ F.
Finally, we prove the theorems.
Proof of Theorem . It follows from (F  ) that F(x, z) ≥  for |z| >  small. By Theorem . for the part λ ∈ (λ A k+δ, λ A k+ ), (GS) λ has a nontrivial solution z λ satisfying Proof of Theorem . With the same argument as above, it follows from Theorem .() and Theorem . for the part λ ∈ (λ A k+ , λ A k+ + δ). We omit the details.
We finally remark that Theorem . is valid for λ ∈ (λ A δ, λ A  ), from which one sees that z =  is a local minimizer of .