Abstract
Keywords:
fourthorder elliptic equation; nontrivial solutions; mountain pass theorem1 Introduction
In this article we study the existence of nontrivial solutions for the fourthorder boundary value problem
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 [13] by using variational methods and fixed point theorems in cones of ordered Banach space with space dimension is one.
On the other hand, The fourorder semilinear elliptic problem
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 [410] 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 fourthorder 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 (PS)condition. Suppose
(1) There exist ρ > 0, α > 0 such that
where B_{p }= {u ∈ E∥u∥ ≤ ρ}.
(2) There is an e ∈ E and ∥e∥ > ρ such that
Then I(u) has a critical value c which can be characterized as
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:
(H1) M(t) is continuous and satisfies
for some m_{0 }> 0. In addition, that there exist m' > m_{0 }and t_{0 }> 0, such that
(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) + bt^{p}, ∀t ∈ R and a.e. x in Ω, where a(x) ∈ L^{q }(Ω), b ∈ R
and if N > 4 and 1 < p < ∞ if N ≤ 4 and ;
(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
Then Equation (1) has at least one nonnegative solution.
Let Ω ⊂ R^{N }be a bounded smooth open domain, be the Hilbert space equipped with the inner product
and the deduced norm
Let λ_{1 }be the positive first eigenvalue of the following second eigenvalue problem
Then from [4], it is clear to see that Λ_{1 }= λ_{1}(λ_{1 } c) is the positive first eigenvalue of the following fourthorder eigenvalue problem
where c < λ_{1}. By Poincare inequality, for all u ∈ H, we have
A function u ∈ H is called a weak solution of Equation (1) if
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
where 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).
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
Since f(x, t) is subcritical by (H3), from the compactness of Sobolev embedding and, following the standard processes we know that to show that I verifies (PS)condition it is enough to prove that {u_{n}} is bounded in H. By contradiction, assume that ∥u_{n}∥ → +∞.
Case I. If is bounded, . We assume that there exist a constant K > 0 such that . By (H1), it is easy to obtain that . Set . Then, from
(H1), (H3), and (H5), we have
On the other hand, it is easy to obtain that
Then, from above, we can have
which contradicts ∥u_{n}∥ → +∞. Therefore {u_{n}} is bounded in H.
Case II. if . By (H1), let l_{2 }= max{1, m'}, we also can obtain that {u_{n}} is bounded in H.
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 with B_{p }= {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
Choosing ε > 0 small enough, we have
by (3), (5), (7) and the Sobolev inequality. So, part 1 is proved if we choose ∥u∥ ρ > 0 small enough.
On the other hand, we have
using (4) and (H5). Hence,
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 conditions are hold:
(H6) f(x, t)t ≥ 0 for x ∈ Ω, t ∈ R;
(H7) , uniformly in a.e x ∈ Ω, where .
Then Equation (1) has at least two nontrivial solutions, one of which is positive and the other is negative.
Let u^{+ }= max{u, 0}, u^{ }= min{u, 0}. Consider the following problem
where
Define the corresponding functional I^{+ }: H → R as follows:
where . Obviously, I^{+ }∈ C^{l}(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
and
where . Obviously, I^{ }∈ C^{1}(H, R). Let u be a critical point of I^{ }which implies that u is the weak solution of Equation (1) with I^{}(u) = I(u).
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 {u_{n}} is bounded in H, by using the Sobolve embedding and the standard procedures, we can get a convergent subsequence. So we need only to show that {u_{n}} is bounded in H.
Let {u_{n}} ⊂ H be a sequence such that
By (H7), it is easy to see that
Now, (9) implies that, for all ϕ ∈ H, we have
Set ϕ = u_{n}, we have
Next, we will show that is bounded. If not, we may assume that ∥u_{n}∥_{L}2 → +∞ as n → +∞. Let , then . From (11), we have
thus {ω_{n}} is bounded in H. Passing to a subsequence, we may assume that there exists ω ∈ H with such that
On the other hand, as n → +∞, by Poincare inequality, it is easy to know that as n → +∞. Thus by (H1), the function . So as n → +∞, by (10), we have
Then ω ∈ H is a weak solution of the equation
The weak maximum principle implies that ω = ω^{+ }≥ 0. Choosing ϕ (x) = φ_{1}(x) > 0, which is the corresponding eigenfunctions of λ_{1}. From (10), we get
On the other hand, we can easily see that Λ = λ_{1}(λ_{1 }+ m') is the eigenvalue of the problem
and the corresponding eigenfunction is still φ_{1}(x). If ω(x) > 0, we also have
which follows that ω ≡ 0 by Λ < β But this conclusion contradicts .
Hence {u_{n}} is bounded in H.
Now we 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 C _{1 }> 0, C_{2 }> 0 such that ∀(x, s) ∈ Ω × R, we have
and
We just prove the case of I^{+}. The arguments for the case of I^{ }are similar. Let ϕ = tφ_{1}. When t is sufficiently large, by (16) and (H1), it is easy to see that
On the other hand, by (17), (H1), the Poincare inequality and the Sobolve embedding, we have
where C_{4 }is a constant. Choosing ∥u∥ = ρ small enough, we can obtain I^{+}(u) ≥ R > 0 if ∥u∥ = ρ.
Proof of Theorem 3.1. From Lemma 3.3, it is easy to see that there exists e ∈ H with ∥e∥ > ρ such that I^{±}(e) < 0.
Define
and
From Lemma 3.3, we have
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 one nontrivial 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 interests
The authors declare that they have no competing interests.
Authors' contributions
Acknowledgements
The authors' would like to thank the referees for valuable comments and suggestions for improving this article.
References

Ma, TF: Existence results for a model of nonlinear beam on elastic bearings. Appl. Math Lett. 13, 11–15 (2000). Publisher Full Text

Ma, TF: Existence results and numerical solutions for a beam equation with nonlinear boundary conditions. Appl Numer Math. 47, 189–196 (2003). Publisher Full Text

Ma, TF: Positive solutions for a nonlocal fourthorder equations of Kirchhoff type. Discrete Contin Dyn Syst (suppl). 694–703 (2007)

An, Y, Liu, R: Existence of nontrivial solutions of an asymptotically linear fourthorder elliptic equations. Nonlinear Anal. 68, 3325–3331 (2008). Publisher Full Text

Bernis, F, Azorero, JG, Peral, I: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourthorder. Adv Diff Equ. 1, 219–240 (1996)

Chen, Y, McKenna, PJ: Traveling waves in a nonlinear suspension beam: theoretical results and numerical observations. J Diff Equ. 135, 325–355 (1997)

Liu, X, Huang, Y: On signchanging solution for a fourthorder asymptotically linear elliptic problem. Nonlinear Anal. 72, 2271–2276 (2010). Publisher Full Text

Lazer, AC, McKenna, PJ: Largeamplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990). Publisher Full Text

McKenna, PJ, Walter, W: Traveling waves in a suspension bridge. SIAM J Appl Math. 50, 703–715 (1990). Publisher Full Text

Pei, R: Multiple solutions for biharmonic equations with asymptotically linear nonlinearities. Bound Value Probl 2010, Article ID 241518. 11, (2010)