CRITICAL GROUPS OF CRITICAL POINTS PRODUCED BY LOCAL LINKING WITH APPLICATIONS

We prove the existence of nontrivial critical points with nontrivial critical groups for functionals with a local linking at 0. Applications to elliptic boundary value problems are given.


Introduction
Let F be a real C 1 function defined on a Banach space X.We say that F has a local linking near the origin if X has a direct sum decomposition X = X 1 ⊕ X 2 with dim X 1 < ∞, F (0) = 0, and, for some r > 0, Then it is clear that 0 is a critical point of F .
The notion of local linking was introduced by Li and Liu [7], [8], who proved the existence of nontrivial critical points under various assumptions on the behavior of F at infinity.These results were recently generalized by Brézis and Nirenberg [3], Li and Willem [9], and several other authors.
In infinite dimensional Morse theory (see Chang [5] or Mawhin and Willem [11]), the local behavior of F near an isolated critical point u 0 , F (u 0 ) = c, is described by the sequence of critical groups where F c is the sublevel set {u ∈ X : F (u) ≤ c}, U is a neighborhood of u 0 such that u 0 is the only critical point of F in F c ∩ U , and H * (•, •) denote the singular relative homology groups.
It was proved in Liu [10] that if F has a local linking near the origin, dim X 1 = j, and 0 is an isolated critical point of F , then C j (F, 0) = 0.In the present paper we use this fact to obtain a nontrivial critical point u with either C j+1 (F, u) = 0 or C j−1 (F, u) = 0.When X is a Hilbert space and F is C 2 , this yields Morse index estimates for u via the Shifting theorem.
When X is a Hilbert space and dF is Lipschitz in a neighborhood of the origin, we extend the result of Liu [10] to the case where F satisfies the "relaxed" local linking condition (see Brézis and Nirenberg [3] and Li and Willem [9]), and thus obtain a nontrivial critical point with a nontrivial critical group in this case also.
We apply our abstract result to elliptic boundary value problems, including an equation asymptotically linear at −∞ and superlinear at +∞, and prove new multiplicity results.

Abstract Result
Throughout this section we assume that F satisfies the Palais-Smale compactness condition (PS) and has only isolated critical values, with each critical value corresponding to a finite number of critical points.Theorem 2.1.Suppose that there is a critical point u 0 of F , F (u 0 ) = c, with C j (F, u 0 ) = 0 for some j ≥ 0 and regular values a, b of F , a < c < b, such that H j (F b , F a ) = 0. Then F has a critical point u with either c < F (u) < b and C j+1 (F, u) = 0, or a < F (u) < c and C j−1 (F, u) = 0. Proof of Theorem 2.1 makes use of the following topological lemma: Proof.Suppose that H j+1 (A , A) = 0. Since H j (A , B ) is also trivial, it follows from the following portion of the exact sequence of the triple (A , A, B ) that H j (A, B ) = 0: Since H j (A, B) = 0, now it follows from the following portion of the exact sequence of the triple (A, B, B ) that H j−1 (B, B ) = 0: Then, since C j (F, u 0 ) = 0, it follows from Chapter I, Theorem 4.2 of Chang [5] that As mentioned before, if F has a local linking near the origin, dim X 1 = j, then C j (F, 0) = 0 (see Liu [10]), and hence the following corollary is immediate from Theorem 2.1: If X is a Hilbert space, F is C 2 , and u is a critical point of F , we denote by m(u) the Morse index of u and by m * (u) = m(u) + dim ker d 2 F (u) the large Morse index of u.We recall that if u is nondegenerate and C q (F, u) = 0, then m(u) = q (see Chapter I, Theorem 4.1 of Chang [5]).Let us also recall that it follows from the Shifting theorem (Chapter I, Theorem 5.4 of Chang [5]) that if u is degenerate, 0 is an isolated point of the spectrum of d 2 F (u), and C q (F, u) = 0, then m(u) ≤ q ≤ m * (u).Hence we have the following corollary: Corollary 2.4.Let X be a Hilbert space and F be C 2 in Theorem 2.1.Assume that for every degenerate critical point u of F , 0 is an isolated point of the spectrum of d 2 F (u). Then F has a critical point u with either Remark 2.5.In particular, Corollary 2.4 yields a critical point u = u 0 with m(u) ≤ j + 1 and j − 1 ≤ m * (u).Benci and Fortunato [2] have proved this fact for the special case where u 0 is a nondegenerate critical point with Morse index j, but without assuming that the critical points of F are isolated.Their proof is based on a generalized Morse theory due to Benci and Giannoni [1].However, Corollary 2.4 says, in addition, that u is at a level different from If X is a Hilbert space and dF is Lipschitz in a neighborhood of the origin, we can relax the local linking condition as in (2).This follows from the following extension of the result of Liu [10] (see also Theorem 5.6 of Kryszewski and Szulkin [6]): Theorem 2.6.Let X be a Hilbert space and dF be Lipschitz in a neighborhood of the origin.Suppose that F satisfies the local linking condition (2), dim X 1 = j.Then C j (F, 0) = 0.
Our proof of Theorem 2.6 uses the following "deformation" lemma: Lemma 2.7.Under the assumptions of Theorem 2.6 there exist a closed ball B centered at the origin and a homeomorphism h of X onto X such that 1. 0 is the only critical point of F in h(B), Proof.Take open balls B , B centered at the origin, with B ⊂ B , such that 0 is the only critical point of F in B and dF is Lipschitz in B , and let B ⊂ B be a closed ball centered at the origin with radius ≤ r (in ( 2)).Since B and (B ) c are disjoint closed sets there is a locally Lipschitz nonnegative function g ≤ 1 satisfying Consider the vector field where P is the orthogonal projection onto X 2 .Clearly V is locally Lipshitz and bounded on X.Consider the flow η(t) = η(t, u) defined by Clearly, η is defined for t ∈ [0, 1].Let h = η(1, •).Since h| (B ) c = id (B ) c and h is one-to-one, h(B) ⊂ B and 1 follows.For u ∈ B ∩ X 2 \{0}, Proof of Theorem 2.6.By 1 of Lemma 2.7, By the local linking condition (2) and 2 and 3 of Lemma 2.7, can also be written as the composition of the inclusion ∂B ∩ X 1 i → B\X 2 and the restriction of h to B\X 2 .Hence we have the following commutative diagram induced by inclusions and h: Since ∂B ∩ X 1 is a strong deformation retract of B \X 2 and h is a homeomorphism, i * and h * are isomorphisms and hence i * is a monomorphism.
Proof.Solutions of (3) are the critical points of the C 2 functional defined on X = H 1 0 (Ω).It is well known that F satisfies (PS).By a standard argument involving a cut-off technique and the strong maximum principle, F has a local minimizer u 0 with 0 < u 0 < a, rank C q (F, u 0 ) = δ q0 .
Let X 1 be the j-dimensional space spanned by the eigenfunctions corresponding to λ 1 , • • • , λ j and let X 2 be its orthogonal complement in X.Then F has a local linking near the origin with respect to the decomposition X = X 1 ⊕ X 2 (see the proof of Theorem 4 in Li and Willem [9]) and hence Also, for α < 0 and |α| sufficiently large, [13]).Therefore, by Theorem 2.1, F has a nontrivial critical point u j with either Since j ≥ 3, a comparison of the critical groups shows that u 0 , u ± 1 , u j are distinct nontrivial critical points of F .
If j ≥ 3, problem (4) has at least three nontrivial solutions.
As in the proof of Theorem 3.1, C j (F, 0) = 0, so, using Lemma 3.3, F also has a nontrivial critical point u j with either C j+1 (F, u j ) = 0 or C j−1 (F, u j ) = 0. Since j ≥ 3, u 0 , u 1 , u j are distinct nontrivial solutions of (4).
and the conclusion follows from Chapter I, Theorem 4.3 and Corollary 4.1 of Chang[5].