This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Existence and multiplicity of solutions for a fourth-order elliptic equation Boundary Value Problems 2012, 2012:6 doi:10.1186/1687-2770-2012-6 Fanglei Wang (wang-fanglei@hotmail.com) Yukun An (anykna@nuaa.edu.cn) ISSN 1687-2770 Article type Research Submission date 26 August 2011 Acceptance date 17 January 2012 Publication date 17 January 2012 Article URL http://www.boundaryvalueproblems.com/content/2012/1/6 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com Boundary Value Problems © 2012 Wang and An ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Existence and multiplicity of solutions for a fourth-order elliptic equation Fanglei Wang ∗1 and Yukun An 2 1 College of Science, Hohai University, Nanjing, 210098, P. R. China 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China ∗ Corresponding author: wang-fanglei@hotmail.com Email address: YA: anykna@nuaa.edu.cn Abstract This article is concerned with the existence and multiplicity of nontrival solutions for a fourth-order elliptic equation        ∆ 2 u − M   Ω |∇u| 2 dx  ∆u = f (x, u), inΩ, u = ∆u = 0, on ∂Ω by using the mountain pass theorem. Keywords: fourth-order elliptic equation; nontrivial solutions; mountain pass theo- rem. 1 1 Introduction In this article we study the existence of nontrivial solutions for the fourth-order boundary value problem        ∆ 2 u − M   Ω |∇u| 2 dx  ∆u = f(x, u), in Ω, u = ∆u = 0, on ∂Ω, (1) where Ω ⊂ R N is a bounded smooth domain, f : Ω × R → R and M : R → R are continuous functions. The existence and multiplicity results for Equation (1) are considered in [1–3] by using variational methods and fixed point theorems in cones of ordered Banach space with space dimension is one. On the other hand, The four-order semilinear elliptic problem        ∆ 2 u + c∆u = f(x, u), in Ω, u = ∆u = 0, on ∂Ω, (2) arises in the study of traveling waves in a suspension bridge, or the study of the static deflection of an elastic plate in a fluid, and has been studied by many authors, see [4–10] and the references therein. Inspired by the above references, the object of this article is to study existence and multiplicity of nontrivial solution of a fourth-order elliptic equation under some conditions on the function M(t) and the nonlinearity. The proof is based on the mountain pass theorem, namely, Lemma 1.1. Let E be a real Banach space, and I ∈ C 1 (E, R) satisfy (P S)- 2 condition. Suppose (1) There exist ρ > 0, α > 0 such that I| ∂B ρ ≥ I(0) + α, where B ρ = {u ∈ E|u ≤ ρ}. (2) There is an e ∈ E and e > ρ such that I(e) ≤ I(0). Then I(u) has a critical value c which can be characterized as C = inf γ∈Γ max u∈γ([0,1]) I(u), where Γ = {γ ∈ C([0, 1], E)|γ(0) = 0, γ(1) = e}. The article is organized as follows: Section 2 is devoted to giving the main result and proving the existence of nontrivial solution of Equation (1). In Section 3, we deal with the multiplicity results of Equation (1) whose nonlinear term is asymptotically linear at both zero and infinity. 2 Main result I Theorem 2.1. Assume the function M(t) and the nonlinearity f(x, t) satisfying the following conditions: 3 (H1) M(t) is continuous and satisfies M(t) > m 0 , ∀ t > 0, (3) for some m 0 > 0. In addition, that there exist m  > m 0 and t 0 > 0, such that M(t) = m  , ∀ t > t 0 . (4) (H2) f(x, t) ∈ C(Ω × R); f(x, t) ≡ 0, ∀x ∈ Ω, t ≤ 0, f(x, t) ≥ 0, ∀x ∈ Ω, t > 0; (H3) |f(x, t)| ≤ a(x) + b|t| p , ∀t ∈ R and a.e. x in Ω, where a(x) ∈ L q (Ω), b ∈ R and 1 < p < N+4 N−4 if N > 4 and 1 < p < ∞ if N ≤ 4 and 1 q + 1 p = 1; (H4) f(x, t) = o(|t|) as t → 0 uniformly for x ∈ Ω ; (H5) There exists a constant Θ > 2 and R > 0, such that ΘF (x, s) ≤ sf(x, s), ∀ | s| ≥ R. Then Equation (1) has at least one nonnegative solution. Let Ω ⊂ R N be a bounded smooth open domain, H = H 2 (Ω)  H 1 0 (Ω) be the Hilbert space equipped with the inner product (u, v) =  Ω (∆u∆v + ∇u∇v)dx, and the deduced norm u 2 =  Ω |∆u| 2 dx +  Ω |∇u| 2 dx. 4 Let λ 1 be the positive first eigenvalue of the following second eigenvalue problem        −∆v = λv, in Ω, v = 0, on ∂Ω. Then from [4], it is clear to see that Λ 1 = λ 1 (λ 1 − c) is the positive first eigenvalue of the following fourth-order eigenvalue problem        ∆ 2 u + cu = λu, in Ω, ∆u = u = 0, on ∂Ω, where c < λ 1 . By Poincare inequality, for all u ∈ H, we have u 2 ≥ Λ 1 u 2 L 2 . (5) A function u ∈ H is called a weak solution of Equation (1) if  Ω ∆u∆vdx + M    Ω |∇u| 2 dx    Ω ∇u∇vdx =  Ω f(x, u)vdx holds for any v ∈ H. In addition, we see that weak solutions of Equation (1) are critical points of the functional I : H → R defined by I(u) = 1 2  Ω |∆u| 2 dx + 1 2  M    Ω |∇u| 2 dx   −  Ω F (x, u)dx, where  M(t) =  t 0 M(s)ds and F (x, t) =  f(x, t)dt. Since M is continuous and f has subcritical growth, the above functional is of class C 1 in H. We shall apply the famous mountain pass theorem to show the existence of a nontrivial critical point of functional I(u). 5 Lemma 2.2. Assume that (H1)–(H5) hold, then I(u) satisfies the (PS)-condition. Proof. Let {u n } ⊂ H be a (PS )-sequence. In particular, {u n } satisfies I(u n ) → C, and I  (u n ), u n  → 0 as n → ∞. (6) Since f(x, t) is sub-critical by (H3), from the compactness of Sobolev embedding and, following the standard processes we know that to show that I verifies (P S)- condition it is enough to prove that {u n } is bounded in H. By contradiction, assume that u n  → +∞. Case I. If  Ω |∇u n | 2 dx is bounded,  Ω |∆u n | 2 dx → +∞. We assume that there exist a constant K > 0 such that  Ω |∇u n | 2 dx ≤ K. By (H1), it is easy to obtain that ˜m = max t∈[0,K] M(t) > m 0 . Set l 1 = min{1, m 0 }, l 2 = max{1, ˜m}. Then, from 6 (H1), (H3), and (H5), we have I(u n ) − l 1 2l 2 I  (u n )u n = 1 2  Ω |∆u n | 2 dx + 1 2  M    Ω |∇u n | 2 dx   −  Ω F (x, u n )dx − l 1 2l 2    Ω |∆u n | 2 dx + M    Ω |∇u n | 2 dx    Ω |∇u n | 2 dx   + l 1 2l 2  Ω f(x, u n )u n dx ≥ 1 2 l 1 u n  2 +  Ω  l 1 2l 2 f(x, u + n )u n − F (x, u + n )  dx ≥ 1 2 l 1 u n  2 +  u n ≥R  l 1 2l 2 f(x, u + n )u + n − F (x, u + n )  dx − C 1 ≥ 1 2 l 1 u n  2 + l 1 2l 2  u n ≥R  f(x, u + n )u + n − 2l 2 l 1 F (x, u + n )  dx − C 1 ≥ 1 2 l 1 u n  2 + l 1 2l 2  u n ≥R  f(x, u + n )u + n − ΘF (x, u + n )  dx − C 1 . On the other hand, it is easy to obtain that I(u n ) − l 1 2l 2 I  (u n )u n ≤ C + Cu n . Then, from above, we can have u n  2 ≤ C + Cu n , which contradicts u n  → +∞. Therefore {u n } is bounded in H. Case II. If  Ω |∇u n | 2 dx → +∞. By (H1), let l 2 = max{1, m  }, we also can ob- tain that {u n } is bounded in H. 7 This lemma is completely proved.  Lemma 2.3. Suppose that (H1)–(H5) hold, then we have (1) there exist constants ρ > 0, α > 0 such that I| ∂B ρ ≥ α with B ρ = {u ∈ H : u ≤ ρ}; (2) I(tϕ 1 ) → −∞ as t → +∞. Proof. By (H1)–(H4), we see that for any ε > 0, there exist constants C 1 > 0, C 2 such that for all (x, s) ∈ Ω × R, one have F (x, s) ≤ 1 2 εs 2 + C 1 s p+1 (7) Choosing ε > 0 small enough, we have I(u) = 1 2  Ω |∆u| 2 dx + 1 2  M    Ω |∇u| 2 dx   −  Ω F (x, u)dx ≥ 1 2  Ω |∆u| 2 dx + 1 2 m 0  Ω |∇u| 2 dx −  Ω F (x, u)dx ≥ 1 2 l 1 u 2 − ε 2 u 2 L 2 − C 1 u p+1 L p+1 ≥ 1 2 (l 1 − ε)u 2 − C 3 u p+1 . by (3), (5), (7) and the Sobolev inequality. So, part 1 is proved if we choose u = ρ > 0 small enough. 8 On the other hand, we have I(u) = 1 2  Ω |∆u| 2 dx + 1 2  M    Ω |∇u| 2 dx   −  Ω F (x, u)dx ≤ 1 2  Ω |∆u| 2 dx + 1 2 m 1  Ω |∇u| 2 dx −  Ω F (x, u)dx ≤ 1 2 l 2 u 2 − u Θ Θ + C 4 . using (4) and (H5). Hence, I(tϕ 1 ) ≤ 1 2 l 2 t 2 ϕ 1  2 − t Θ ϕ 1  Θ Θ + C 4 → −∞ as t → +∞ and part 2 is proved.  Proof of Theorem 2.1. From Lemmas 2.2 and 2.3, it is clear to see that I(u) satisfies the hypotheses of Lemma 1.1. Therefore I(u) has a critical point.  3 Existence result II Theorem 3.1. Assume that (H1) holds. In addition, assume the following condi- tions are hold: (H6) f(x, t)t ≥ 0 for x ∈ Ω, t ∈ R; (H7) lim t→0 f(x,t) t = α, lim |t|→+∞ f(x,t) t = β, uniformly in a.e x ∈ Ω, where 0 ≤ α min{1,m 0 } < λ 1 (λ 1 + m  ) < β < +∞. Then Equation (1) has at least two nontrivial solutions, one of which is positive and the other is negative. 9 [...]... referees for valuable comments and suggestions for improving this article References [1] Ma, TF: Existence results for a model of nonlinear beam on elastic bearings Appl Math Lett 13, 11–15 (2000) [2] Ma, TF: Existence results and numerical solutions for a beam equation with nonlinear boundary conditions Appl Numer Math 47, 189–196 (2003) [3] Ma, TF: Positive solutions for a nonlocal fourth-order equations... that they have no competing interests Authors’ contribution In this manuscript the authors studied the existence and multiplicity of solutions for an interesting fourth-order elliptic equation by using the famous mountain pass lemma Moreover, in this work, the authors’ supplements done in [1–3] All authors 15 typed, read and approved the final manuscript Acknowledgment The authors’ would like to thank... nontrival critical point in H corresponding to this value This critical in nonnegative, then the strong maximum principle implies that is a positive solution of Equation (1) By an analogous way we know there exists at least one negative solution, which is a nontrivial critical point of I − Hence, Equation (1) admits at least a positive solution and a negative solution Competing interest The authors’ declare... 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Let u be a critical point of I − which implies that u is the weak solution of Equation (1) with I − (u) = I(u) 10 Lemma 3.2 Assume that (H1), (H6), and (H7) hold, then I ± satisfies the (PS) condition Proof We just prove the case of I + The arguments for the case of I − are similar Since Ω is bounded and (H7) holds, then if {un } is bounded in H, by using the Sobolve embedding and the standard procedures,... is easy to see that there exists e ∈ H with e > ρ such that I ± (e) < 0 Define P = {γ : [0, 1] → H : γ is continuous and γ(0) = 0, γ(1) = e}, 14 and c± = inf max I ± (γ(t)) γ∈P t∈[0,1] From Lemma 3.3, we have I ± (0) = 0, I ± (e) < 0, I ± (u)|∂Bρ ≥ R > 0 Moreover, by Lemma 3.2, the functions I ± satisfies the (PS)-condition By Lemma 1.1, we know that c+ is a critical value of I + and there is at least... ∀u ∈ H, Ω f + (x, t)dt Obviously, I + ∈ C 1 (H, R) Let u be a critical point of I + which implies that u is the weak solution of Equation (8) Futhermore, by the weak maximum principle it follows that u ≥ 0 in Ω Thus u is also a solution of Equation (1) Similarly, we also can define f − (x, t) =    f (x, t),  if t ≤ 0,    0, if t > 0, and  I − (u) = 1 2  1 |∆u|2 dx + M  2 Ω where F − (x, u)... oscillations in suspension bridge: some new connections with nonlinear analysis SIAM Rev 32, 537–578 (1990) 16 [9] McKenna, PJ, Walter, W: Traveling waves in a suspension bridge SIAM J Appl Math 50, 703–715 (1990) [10] Pei, R: Multiple solutions for biharmonic equations with asymptotically linear nonlinearities Bound Value Probl 2010, Article ID 241518, 11 (2010) 17 ... prove that the functionals I ± has a mountain pass geometry Lemma 3.3 Assume that (H1), (H7) hold, then we have (1) there exists ρ, R > 0 such that I ± (u) > R, if u = ρ; (2) I ± (u) are unbounded from below Proof By (H7), for any ε > 0, there exists C1 > 0, C2 > 0 such that ∀(x, s) ∈ Ω×R, we have 1 F (x, s) ≤ (α + ε)s2 + C1 sp+1 2 (15) 1 F (x, s) ≥ (β − ε)s2 − C2 , 2 (16) and     where 2 < p < 2∗ . author: wang-fanglei@hotmail.com Email address: YA: anykna@nuaa.edu.cn Abstract This article is concerned with the existence and multiplicity of nontrival solutions for a fourth-order elliptic equation        ∆ 2 u. Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Existence and multiplicity of solutions for a. distribution, and reproduction in any medium, provided the original work is properly cited. Existence and multiplicity of solutions for a fourth-order elliptic equation Fanglei Wang ∗1 and Yukun An 2 1 College