EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS FOR A SEMILINEAR BIHARMONIC EQUATION WITH WEIGHT FUNCTIONS

. We consider the existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with the concave-convex nonlinearities f ( x ) | u | q − 1 u and h ( x ) | u | p − 1 u under certain conditions on f ( x ) , h ( x ), p and q . Applying the Nehari manifold method along with the ﬁbering maps and the minimization method, we study the eﬀect of f ( x ) and h ( x ) on the existence and multiplicity of nontrivial solutions for the semilinear biharmonic equation. When h ( x ) + (cid:54) = 0, we prove that the equation has at least one nontrivial solution if f ( x ) + = 0 and that the equation has at least two nontrivial solutions if (cid:82) Ω | f + | r d x ∈ (0 ,Λ r ), where r and Λ are explicit numbers. These results are novel, which improve and extend the existing results in the literature.


XIYOU CHENG, ZHAOSHENG FENG AND LEI WEI
where Ω is a bounded smooth domain in R N (N 3), g λ : Ω×R → R and parameter λ ∈ R + 0 R + 0 = (0, +∞) , see [2-4, 6, 13, 16, 18, 24] and the references therein. As g λ is sublinear, for example, g λ = λu q , 0 < q < 1, the monotone iteration scheme or the upper and lower solution method are effective, see [12]. As g λ is superlinear and subcritical, for instance, g λ = λu + |u| p−1 u, 1 < p < N +2 N −2 , variational methods can be applied, see [23]. In contrast to the sublinear case and the superlinear case, Ambrosetti-Brezis-Cerami [3] considered the problem (2) in the case that g λ is, roughly, the sum of a sublinear term and a superlinear term. In precise, they dealt with the problem −∆u = λu q + u p , in Ω, where 0 < q < 1 < p N +2 N −2 , and showed that the problem (3) admits at least two positive solutions for λ ∈ R + 0 small enough. Sun-Li [16] considered a similar problem −∆u = u q + λu p , in Ω, 0 u ∈ H 1 0 (Ω), with 0 < q < 1 < p = N +2 N −2 . They investigated the value of Λ-the supremum of the set λ associated with the existence and multiplicity of positive solutions and established the uniform lower bounds for Λ. Wu [24] considered the subcritical case of the problem (3) with λu q being replaced by λf (x)u q , where f (x) ∈ C(Ω) is a sign-changing function, and proved that the problem (3) has at least two positive solutions as λ ∈ R + 0 is sufficiently small. Some interesting extensions of (3) have been carried out in the framework of quasi-linear elliptic equations or systems, semilinear second-order elliptic systems and semilinear fourth-order elliptic equations, see [9,11,15,20,25,26]. For instance, Yang-Wang [25] showed that there exists at least two positive solutions when λ ∈ R + 0 is small enough for the semilinear biharmonic elliptic problem where Ω is a bounded smooth domain in R N (N > 4), 0 < q < 1 < p < N +4 N −4 , λ ∈ R + 0 , and f (x) ∈ C(Ω, R) and h(x) ∈ C(Ω, R + 0 ) are weight functions. In the present paper, our goal is to study the existence and multiplicity of nontrivial solutions for the problem (1). For simplicity and convenience, we denote the norm of u in L s (Ω) by |u| s = ( Ω |u(x)| s ) 1/s and the norm of u in C(Ω) by |u| ∞ = max x∈Ω |u(x)|. We also denote H 1 0 (Ω) ∩ H 2 (Ω) by H(Ω), endowed with the norm u = |∆u| 2 . Let S be the best Sobolev constant for the embedding of H(Ω) in L p+1 (Ω), precisely, |u| p+1 S u for all u ∈ H(Ω) (see [1]).
We define It is well-known that the nontrivial weak solutions of the problem (1) are the nonzero critical points of the energy functional J (see [5,14]). In order to find the nonzero critical points of the functional J, we focus on the Nehari minimization problem: We split N into three parts Note that all nonzero solutions of (1) are obviously in the Nehari Manifold-N. Hence, our approach to solve the problem (1) is to equivalent to analyzing the structure of N, and the minimization problem for J on N + and N − by applying the Nehari manifold method along with the fibering maps and the minimization method.

Remark 2.
As pointed out in [8], we do not claim in the statements of Theorems 1.1 and 1.2 that the solutions are nonnegative. The reason is that we can not use, for a local minimizer u ∈ H(Ω), the relation J(u) = J(|u|) to deduce nonnegativity of solution, since it can happen that |u| ∈ H(Ω) even if u ∈ H(Ω).
The paper is organized as follows. In Section 2, we present some technical lemmas. In Section 3, we prove Theorems 1.1 and 1.2, respectively.

2.
Preliminaries. In this section, we present several lemmas, which will be used in the proofs of our main results. and By (7)- (8), it follows from Sobolev's inequality and Hölder's inequality that and Using (9) and (10) yields Hence, by (11) we arrive at the desired result.
The following lemma shows that the local minimizers on N are the critical points for J.
Proof. If u 0 is a local minimizer of J on N, then u 0 is a solution of the optimization problem minimize J(u) subject to ψ(u) = 0. By using the Lagrange multipliers, there exists θ ∈ R such that Thus, we get From u 0 ∈ N and Lemma 2.1, we have J (u 0 ), u 0 = 0 and ψ (u 0 ), u 0 = 0. Thus θ = 0 due to (13), which together with (12) implies that J (u 0 ) = 0 in [H(Ω)] * . Lemma 2.3. The following two statements are true.
Proof. (i) In view of (6) and u ∈ N + , we have (ii) In view of (6) and u ∈ N − , we get Then, we have the following lemma.
Proof. For u ∈ H(Ω)\{0}, we set It is easy to see that This implies that t − u ∈ N − . By a direct calculation, we have Clearly, dJ(tu) dt > 0 for t ∈ (0, t − ) and dJ(tu) dt < 0 for t ∈ (t − , +∞). Hence, J(tu) attains its maximum at t − , that is, (15) and there exist unique t + and t − such that 0 < t + < t max < t − , and and Similar to (i), we have t + u ∈ N + , t − u ∈ N − , and If Ω h(x)|u| p+1 dx 0 and Ω f (x)|u| q+1 dx > 0, by the monotonous increase of s(t) with s(0) = 0 and lim t→+∞ s(t) = +∞, there is a unique t + ∈ (0, +∞) such that Let us consider the existence of the least energy (or ground state) solutions for the semilinear biharmonic problem where Ω ⊂ R N (N > 4) is an open bounded domain with the smooth boundary, the weight function f ∈ C(Ω) and q ∈ (0, 1). The energy functional is given by and the corresponding Nehari minimization problem is Then, we have the following result.
Proof. We separate the proof into the following steps.
Step 1. We claim that M f = ∅. Let Then φ ∈ L ∞ (Ω), and there exists a sequence of the smooth functions {φ n } ∞ n=1 ⊂ C ∞ 0 (Ω) such that φ n → φ (as n → ∞) in L p+1 (Ω). It follows from Minkowski's and Hölder's inequalities that , where f ± = max{±f, 0}, which implies that Note that f + = 0 and So, Ω f + |φ| q+1 dx ∈ R + 0 . We now choose n ∈ N large enough such that We consider It is clear that there exists a unique t n ∈ R + 0 such that L(t n ) = 0, that is, Step 2. We show that E f is bounded below on H(Ω) and γ f < 0.
By using Hölder's, Sobolev's and Young's inequalities, we have Thus, E f is bounded below on H(Ω) and E f is bounded below on M f . Hence, Step 3. We prove the existence of the minimizers for E f on M f . For w ∈ M f , it follows from Hölder's and Sobolev's inequalities that Let {w n } be a minimizing sequence for E f on M f . By (17) and the compact imbedding theorem, there exist a subsequence still denoted by {w n } and w f in H(Ω) such that w n w f weakly in H(Ω) and w n → w f strongly in L p+1 (Ω).

This yields a contradiction with
We now show that w n → w f strongly in H(Ω). Otherwise, w f < lim inf n→∞ w n . That is, This is a contradiction. Hence, w n → w f strongly in H(Ω). This implies w f ∈ M f and Step 4. We show the existence of nonzero weak solutions of the problem (16).
Thus, by the theory of Lagrange multipliers, there exists ∈ R such that Then Since Ψ f (w 0 ), w 0 = (1 − q) w 0 2 > 0, we have = 0. Hence, E f (w 0 ) = 0. In other words, a minimizer of E f on M f is a nonzero critical point of E f .
Noticing that E f (w f ) = γ f and w f ∈ M f , we obtain that w f is a nonzero weak solution of the problem (16). Lemma 2.6. J is coercive and bounded below on N.
Proof. If u ∈ N, by Hölder's and Young's inequalities, it follows from (5)-(6) that Thus, J is coercive and bounded below on N.
As a conclusion in this section, we give the similar results to [25, Lemmas 2.8-2.9] whose proofs can be processed based on [19] in a straightforward way, so we omit them.
[25] For any u ∈ N − , there exist > 0 and a differentiable functional 3. Proof of Theorems 1.1-1.2. We start with the existence of the minimizing sequences for J on N and N − as |f + | r is sufficiently small.
This implies That is, To show that J (u n ), ϕ → 0 as n → ∞, ϕ ∈ H(Ω), This yields a contradiction to (38). Hence, we have (ii) According to Lemma 2.6 and the Ekeland variational principle, there is a minimizing sequence {u n } ⊂ N − such that and In view of {u n } ⊂ N − , we have Using Hölder's, Sobolev's and Young's inequalities, it follows from (40) that there From (41)-(43), processing an analogous argument as we did in the proof of Then by Lemma 2.6 and the compact imbedding theorem, there exist a subsequence {u n } and u + 0 ∈ H(Ω) such that u n u + 0 weakly in H(Ω), and We make a claim that If not, by (47) there holds Combining {u n } ⊂ N with (46) and (48) leads to This contradicts lim n→∞ J(u n ) = α < 0.
In combination with (44)-(47), it is easy to verify that u + 0 ∈ N is a nontrivial weak solution of the problem (1). Now we prove that u n → u + 0 strongly in H(Ω). Suppose the contrary. Then u + 0 < lim inf n→∞ u n and This contradicts u + 0 ∈ N. Hence, u n → u + 0 strongly in H(Ω). This implies J(u n ) → J(u + 0 ) = α as n → ∞.
(ii) By Lemma 2.2 and Theorem 3.2 (i), u + 0 is a nontrivial weak solution to the problem (1).
Next, we establish the existence of a local minimum for J on N − . Then by Lemma 2.6 and the compact imbedding theorem, there exist a subsequence {u n } and u − 0 ∈ H(Ω) such that u n u − 0 weakly in H(Ω), u n → u − 0 strongly in L p+1 (Ω) and u n → u − 0 strongly in L q+1 (Ω). Due to the fact that J (u − 0 ), ϕ = lim n→∞ J (u n ), ϕ = 0 for all ϕ ∈ H(Ω), we can see that u − 0 ∈ N is a nontrivial weak solution of the problem (1). To prove that u n → u − 0 strongly in H(Ω), we suppose the contrary that u − 0 < lim inf n→∞ u n and then This contradicts the fact u − 0 ∈ N. Hence, u n → u − 0 strongly in H(Ω). This implies J(u n ) → J(u − 0 ) = α − as n → ∞. In view of N 0 = ∅ (see Lemma 2.1) and {u n } ⊂ N − , we obtain u − 0 ∈ N − . (ii) It follows from Lemma 2.2 and Theorem 3.3 (i) that u − 0 is a nontrivial weak solution to the problem (1).