Multiple sign-changing solutions of an elliptic eigenvalue problem

We prove 
the existences of multiple sign-changing solutions for a 
semilinear elliptic eigenvalue problem with constraint by using 
variational methods under weaker conditions.


Introduction
Multiple solutions for elliptic eigenvalue problem have been studied in some papers such as [7] [9], but they did not locate whether those solutions were sign-changing or not. [5] Theorem 4.4 got multiple sign-changing solutions under the assumption that the functional was of C 2 . Both [6] and [8 Theorem 2] gave a result on existence of three solutions with one sign-changing for (4.1) with f being not odd and obtained multiple sign-changing solutions under further condition. In this paper, we will obtain multiple sign-changing solutions and a pair of non sign-changing solutions of (4.1) under weaker conditions. Our result improves that of [5,6,7,8,9] and so on.
The paper is organized as follows. Section 2 contains some preliminary technical results. We construct a new pseudo-gradient flow of a C 1 functional, and prove the deformation lemma for their own sake. Section 3 is devoted to prove an abstract theory, which asserts multiplicity of sign-changing solutions. In fact, our theory is not confined to situations of cone-structures, we may consider the locations of critical points in relation to some given invariant sets of the flow. In section 4, we apply our theory to a nonlinear elliptic eigenvalue problem with constraint, establishing multiple sign-changing solutions under weaker conditions than those of correlated references.

Preliminaries
Lemma 2.1 Assume that E is a Hilbert space, given r > 0, S r = {u ∈ E : u 2 = r 2 }, J ∈ C 1 (S r , R), J (u) = −a(u)u − A(u) for u ∈ S r , where a(u) = <−A(u),u> is an open covering of E 0 , then N u possesses a locally finite refinement which will be denoted by {U λ : λ ∈ Λ}. There exists a locally finite Then B : E 0 → S r is locally Lipschitz continuous as a consequence of the Lipschitz continuity of β λ , and the locally finiteness of U λ , so does W (u) : We claim W is also Lipschitz continuous. Indeed, Since A(u) is bounded in S r , so are B(u) and b(u). Thus W is also Lipschitz continuous. For any u ∈ E 0 , there are only a finite number of U λ denoted by U λ i (i = 1, 2, · · · , n(u)), which contain u. For each λ i , there exists U λi ∈ E 0 such that From this inequality, we easily get which imply W is a pseudo-gradient vector field for J on S r \K. Finally, if J is even and M = −M , we enlarge the covering {U λ : λ ∈ Λ} by adding the sets {−U λ : λ ∈ Λ}. Then the vector V (u) = 1 2 (W (u) − W (−u)) is odd, locally Lipschitz continuous and satisfies (2.1)(2.2).
Consider the pseudo-gradient flow σ on E associated to the vector field W : Let E be a Hilbert space and X ⊂ E a Banach space densely embedded in E. Assume that E has a closed convex cone P E and that P := P E X has interior points in X, i.e., P = intP ∂P , with intP the interior and ∂P the boundary of P in X.
We use the following notation: Let · and · X denote the norms in E and X respectively. We use d E (·, ·) and d X (·, ·) to denote the distance in E and X, S r = X S r .
The first assumption we make is: Lemma 2.2 [5] Let Ψ : E → R be a locally Lipschitz continuous function. Then Ψ X : X → R is also locally Lipschitz continuous in the topology of X.
Then there is ε 0 > 0, such that for any 0 < ε < ε 0 and any compact subset (iv) J(η(·, u)) is nonincreasing for any u ∈ S r ; (v) η(t, M ) ⊂ M for any t ∈ [0, 1]; (vi) η(t, ·) is odd, if J is even and M is symmetric about the origin. Proof. Due to the (P S) condition, we can choose ε 0 > 0, such that and The followings are similar to the proof of Lemma 2.4 [5], in which X is replaced by S r , and the gradient vector ∇φ(u) is replaced by the pseudo gradient vector W (u). We omit them.

Remark 2.4
It is easy to check that both the union and intersection of a finite number of admissible invariant sets for J are still admissible invariant sets for J.
3. An abstract theory and its proof First, we need the notion of genus and its properties (see [7]). Let For preciseness, we denote i X (A) and i E (A) to be the genus of A in S r and S r respectively.
Proposition 3.1 Assume that A, B ∈ Σ X , h ∈ C( S r , S r ) is an odd homeomorphism. Then: This proposition is still true when we replace Σ X by Σ E with obvious modification. We shall use the notion S = S r \(P −P ).
Theorem 3.1 Let J ∈ C 1 (S r , R) and (Φ) hold. Assume that J is even, bounded from below and satisfies (P S) c condition for all c = 0. Let P S r ⊂ X be an admissible invariant set for J. Assume K(J) ∂P = ∅. Then J has infinitely many distinct pairs of critical points in S r \(P −P ). Proof. That J possesses infinitely many pairs of critical points follow from e.g. [7,Theorem 8.10]. The question is whether they belong to P and −P . We now rule this out.

AIXIA QIAN AND SHUJIE LI
and for m ≥ 2 define Since J is bounded from below, Lemma 3.1 implies We claim that for m = 2, 3, · · · , Moreover (P S) condition implies K c m ( S r \(P −P )) is a compact set. We also claim that if, for l ≥ 1 and k ≥ 2, Please see the proof of Theorem 2.1 [5] for those of (3.1) (3.2) in detail. Then there are infinitely many pairs of critical points in S r \(P −P ).

Application
As application of our abstract theory, we consider a nonlinear eigenvalue problem: for r > 0 fixed, where, Ω ⊂ R N is a smooth bounded domain. We want to find solution of the form (λ, u). We say (λ, u) a positive solution if u is positive, a negative solution if u negative, and a sign-changing solution if u sign-changing. We refer the following assumptions.
where,<, > is the inner product in E given by . From (f 2 ), we have < Ψ (u), u > = 0 for u ∈ S r and know that there is a one-toone correspondence between critical point of C 1 functional J and weak solution of (4.1). Thus, if u is a critical point of J, then (λ, u) is a weak solution of (4.1) with On E, let us define P E = {u ∈ E : u(x) ≥ 0, a.e. in Ω}, which is a closed convex cone. Set P = P E X, then P is a closed convex cone in X. Furthermore, P = intP ∂P under the topology of X, i.e., there exist interior points in X. We may define a partial order relation: Proof. By the standard elliptic regular theory, Proof. If not, there exists a λ ∈ Λ with a λ E ≥ r + 3 such that β λ (v) = 0 for some v ∈ ℵ. Then suppβ λ (v) ⊂ B(a λ , r a λ + 1) ⊂ E\B(0, r + 1), while v ∈ S r , a contradiction.

Remark 4.2 From Proposition 4.2, it follows that
Proposition 4.3 Let X 0 = W 2,q 0 (Ω) for some q ≥ 2 large such that the embedding from X 0 into X = C 1 0 (Ω) is compact. It is well known that there exist a finite sequence of Banach spaces X 1 , X 2 , · · · , X m , such that X 0 → X 1 → · · · → X m and the embeddings from X k into X m are compact for k = 0, 1, · · · , m − 1, A(X k ) ⊂ X k−1 with 1 ≤ k ≤ m maps bounded sets to bounded sets. Then B(X k ) ⊂ X k−1 maps bounded sets to bounded sets.

Proof. Remark 2.1 and
Since the embedding from X m−1 into X m is compact, ℵ is compact in X m and for all v ∈ ℵ, there exists a neighborhood U v of v, such that U v intersects with finitely many sets of {U λ : λ ∈ Λ}. Then there exist finitely many β λ ∈ {β λ : λ ∈ Λ} such that suppβ be the open covering of ℵ under the topology of X m , then there are finitely many coverings U v1 , · · · , U vn of }, then for all v ∈ ℵ and for all β λ ∈ {β λ : λ ∈ Λ}\ , β λ (v) = 0.
Similarly, we see that ℵ is bounded in X j , for j = 0, 1, · · · , m − 3. Proof. The proof is similar to that of Lemma 1.6 [6]. Firstly, by the strong maximum principle, we know that if u ∈ P \{0}, then A(u) ∈ intP , so does B(u).
Secondly, ∀u ∈ P , since B(u) ∈ P , we can choose δ > 0 small such that if δ > h > 0, we have Brezis-Martin theorem implies σ(t, u) ∈ P for all t ∈ [0, T (u)). It follows that Let w(t) = − t 0 b σ(s, u 0 ) ds, we have w (t) > 0, w(t) > 0 and w(t) is strictly increasing. Denote the inverse function of w(t) by w −1 (t). From (4.4), for u 0 ∈ P S r , we have 1 Note that F t is a compact set in X. It follows from (4.5) that F t ∈ intP and hence CoF t ∈ intP , where CoF t is the closed convex hull of F t in X. Note that Finally, it follows (4.2) that As the same proof above, we have  Proof. [7,Theorem 8.10] implies that J satisfies (P S) condition. The requirement (a) is satisfied automatically. Remark 4.3 implies that P S r is an invariant set of the negative gradient flow of J, and (d) is satisfied. To prove (b), let u n = σ(t n , v) for some v ∈ S r \ P (−P ) , and let t n → ∞ be a sequence such that u n → u in S r for some u ∈ K(J), then σ(t, v) is bounded in X m for t ≥ 0. Proposition 4.3 implies σ(t, v) for t ≥ 0 is bounded in X 0 . Since the embedding from X 0 into X is compact, u n = σ(t n , v) → u in X. For (c), if u n ∈ K(J) P (−P ) , u n → u in S r , then u n is bounded in E. By condition (f 1 ), the elliptic theory and the bootstrap argument, we get u n is bounded in W 2,q 0 (Ω) for some q large such that the embedding from W 2,q 0 (Ω) into X is compact, thus u n → u in S r . Theorem 4.1 Under (f 1,2,3 ), (4.1) has infinitely many sign-changing solutions.
Proof. Under these hypothesis, we know that Ψ(u) < 0 for any u ∈ S r . As is proved in [7] that J is bounded from below and satisfies (P S) c for all c < 0. By the maximum principle K(J) ∂P = ∅. It follows proposition 4.5 that P S r is an admissible invariant set for J. Then Theorem 3.1 implies the result. Proof. Please see Theorem 1 [8] for the proof in detail.