ON THE EXISTENCE OF NONTRIVIAL SOLUTIONS FOR A FOURTH-ORDER SEMILINEAR ELLIPTIC PROBLEM

where, 2 is the biharmonic operator and Ω is a bounded domain in Rn with smooth boundary ∂Ω. This fourth-order semilinear elliptic problem can be considered as an analogue of a class of second order problems which have been studied by many authors. In [3], there was a survey of results obtained in this direction. Known results about (P) were concerned with the case c < λ1 the first eigenvalue of − in H 0 (Ω). In [8], the author proved the existence of a negative solution of (P) by a degree theory with f (x,u) = b[(u+ 1) − 1]. [4] showed that there existed multiple solutions for f (x,u)= bg(x,u) by using variational approach. Our recent work obtained a positive solution and a negative solution of (P) by Mountain Pass Theorem, and one more nontrivial solution by Morse theory. It is natural to ask what additional phenomena if c goes beyond λ1. In [5], the author considered the problem (P) with f (x,u)= bg(x,u), and got two solutions by using a “variation of linking” theorem under certain conditions. In the present work, we study the problem (P) with c ≥ λ1 by using variational approach. In Section 2, we prove the existence of one nontrivial solution by Linking Theorem including the Saddle Point Theorem, whether c is one of the eigenvalues λk of (− ,H 0 (Ω)) or not. In Section 3, we obtain two nontrivial solutions by using a “variation of linking” theorem. Section 4 is devoted to prove the multiplicity of nontrivial solutions, by using the pseudo-index theory. Of course, our results are still valid for second-order semilinear elliptic problem under weaker conditions.


Introduction
Let us consider the problem where, 2 is the biharmonic operator and Ω is a bounded domain in R n with smooth boundary ∂Ω.This fourth-order semilinear elliptic problem can be considered as an analogue of a class of second order problems which have been studied by many authors.In [3], there was a survey of results obtained in this direction.Known results about (P) were concerned with the case c < λ 1 the first eigenvalue of − in H 1 0 (Ω).In [8], the author proved the existence of a negative solution of (P) by a degree theory with f (x,u) = b[(u + 1) + − 1].[4] showed that there existed multiple solutions for f (x,u) = bg(x,u) by using variational approach.Our recent work obtained a positive solution and a negative solution of (P) by Mountain Pass Theorem, and one more nontrivial solution by Morse theory.It is natural to ask what additional phenomena if c goes beyond λ 1 .In [5], the author considered the problem (P) with f (x,u) = bg(x,u), and got two solutions by using a "variation of linking" theorem under certain conditions.In the present work, we study the problem (P) with c ≥ λ 1 by using variational approach.
In Section 2, we prove the existence of one nontrivial solution by Linking Theorem including the Saddle Point Theorem, whether c is one of the eigenvalues λ k of (− ,H 1 0 (Ω)) or not.In Section 3, we obtain two nontrivial solutions by using a "variation of linking" theorem.Section 4 is devoted to prove the multiplicity of nontrivial solutions, by using the pseudo-index theory.Of course, our results are still valid for second-order semilinear elliptic problem under weaker conditions.

The existence of one nontrivial solution
, and each eigenvalue is repeated according to its multiplicity.Let e k be the eigenfunction corresponding to λ k orthogonal in L 2 (Ω), we can choose e 1 > 0 in Ω.
where, F(x,t) be taken as a norm on H, one can use the Mountain Pass theorem to establish the existence of a weak solution of (P) (and even a positive solution).However, if c ≥ λ 1 , our previous mechanism fails, we will apply the Linking Theorem to obtain the weak solution of (P).
Proof.First of all, we observe that where, i * : It is enough to prove that ( u n ) n∈N is bounded, because of (2.8) and (f 1 ).We consider the case N ≥ 5. Form (f 2 ), we obtain the existence c 1 > 0 such that (2.9) Let β ∈ (α −1 ,2 −1 ), for n large enough and c 2 ,c 3 > 0, we have where, It is easy to verify that (u n ) is bounded in H using the fact that dimY is finite.
A standard argument shows that {u n } has a convergent subsequence in H. Therefore, J satisfies the (PS) condition.
Proof.(1) We consider the case N ≥ 5. We will verify the assumptions of the Linking Theorem.The (PS) condition follows form the preceding Lemma 2.1.
(2) By (f 1 ) (f 2 ), we have On Z, we obtain (2.12) 676 Solutions for a fourth-order semilinear elliptic problem By Sobolev imbedding theorem, there exists r > 0, such that (2.13) (3) By (f 4 ), on Y we have Define z := re n+1 / e n+1 H .It follows (2.9), for u = y + λz with λ > 0, we deduce (2.15) Since on the finite dimensional space Y Rz all norms are equivalent, then we get (2.16) Thus, there exists ρ > r such that max where M 0 is as above.By the Linking Theorem, there exists a critical point u of J satisfying J c (u) ≥ b > 0. Since J c (0) = 0, then u is a nontrivial solution of (P).Proof.Please see [6, Theorem 2.15] for its proof in detail, where Similarly, we can obtain the following corollary.

problem (P) has a positive solution and a negative solution.
A. Qian and S. Li 677 Proof.By the truncation technique and the Mountain Pass theorem, the problem has a solution u ≡ 0 satisfying
From (f 4 ) , further using the strong maximum principle [2], we deduce u > 0, that is, u is the positive solution of (P).
Proof.Condition (f 4 ) is stronger than (f 4 ), which is also applied to show that, for As the similar proof of Theorem 2.2, we obtain the result.
Remark 2.7.In Corollary 2.5, we obtain a positive solution of (P) by using truncation technique, if c < λ 1 .However, if c ≥ λ 1 , we cannot expect a positive solution of (P).Indeed, if v 1 is the eigenfunction corresponding to λ 1 , we can assume v 1 > 0 in Ω.Therefore, if u is a solution of (P), we get If u is positive in Ω, the left-hand side of (2.22) is nonnegative by (f 4 ) , while the righthand side is nonpositive, since c ≥ λ 1 .Thus, there can only be a positive solution u(x) if c = λ 1 , and p(x,u(x)) ≡ 0. If c = λ k < λ k+1 , we can apply the Saddle Point Theorem to obtain a nontrivial solution of (P).[6,Theorem 4.6].Let E = V X, where E is a real Banach space and V ≡ {0} is finite dimensional.Suppose J ∈ C 1 (E,R) satisfies (PS) condition, and (I 1 ) there is a constant α and a bounded neighborhood D of 0 in V , such that J| ∂D ≤ α, (I 2 ) there is a constant β > α such that J| X ≥ β.

Saddle Point Theorem
Then J possessed a critical point whose critical value c ≥ β.
Theorem 2.8.Under the following conditions

problem (P) possesses a nontrivial solution.
Proof.Since (i), J c is of C 1 .Let V := span{e 1 ,...,e k }, and X := span{e j | j ≥ k + 1}, so X = V ⊥ .Therefore H = V X.We will show that J c satisfies (i) (ii) and (PS) condition.Then our result follows from the Saddle Point Theorem.
for all u ∈ H via the Hölder and Poincaré inequality.On X, the norms u 2 = Ω (| u| 2 − c|∇u| 2 ) and u 2 H are equivalent, we have which shows J c is bounded from below on X, that is, (I 2 ) holds.Next, if u ∈ V , then u = u 0 + u − , where u 0 ∈ E 0 := span{e j |λ j = c}, and u − ∈ E − := span{e j |λ j < c}.Then (2.25) Estimating the last term as in (2.23), since all norms are equivalent on the finite dimensional subspace E − , we have Now, (2.26) and (ii) show J c (u) → −∞ as u → ∞ in V .Hence J c satisfies (I 1 ).
A. Qian and S. Li 679 Lastly to verify (PS) condition, it suffices to show that Consequently, since X = V ⊥ , by (2.27) and an estimate like (2.23), we get ) is bounded in H and we are through.Indeed, By what has already been shown, the first term on the right is bounded independently of m.Therefore so ( Ω F(x,u 0 m )dx) is bounded, which implies (u 0 m ) is bounded as the proof of Lemma 4.21 [6].

The existence of two nontrivial solution
By using the following "a variation of linking" theorem, we can obtain at least two solutions of (P).Theorem 3.1 ("a variation of linking") [7,Corollary 2.4].Let N be a subspace of a Hilbert space H, such that 0 < dimN < ∞, and M = N ⊥ .Assume J is a continuously differentiable functional on H, which satisfies for some α < β, 0 < δ < R and w 0 ∈ M\{0}, If J satisfies the (PS) condition, then there are at least two solutions of J (u) = 0, one satisfies J(u) ≤ α and the other J(u) ≥ β.
thus, there exist ρ, α 0 > 0 small enough, such that (II) For m ≥ 1 fixed, since all norms are equivalent on the finite dimensional space Y j+m , by (2.9) there exists a sufficiently big constant R > ρ, such that and the (PS) condition is obtained by Lemma 2.1.Therefore, Theorem 4.2 implies that J c admits at least m distinct pairs of critical points.Since m is arbitrary and λ j → +∞ as j → ∞, then (P) has infinitely many nontrivial solutions.