Abstract
The existence of multiple solutions for a class of fourthorder elliptic equation with respect to the generalized asymptotically linear conditions is established by using the minimax method and Morse theory.
Keywords:
fourthorder elliptic boundary value problems; multiple solutions; mountain pass theorem; Morse theory.1 Introduction
Consider the following Navier boundary value problem
where Δ^{2 }is the biharmonic operator, and Ω is a bounded smooth domain in ℝ^{N }(N > 4), and the first eigenvalue of Δ in .
The conditions imposed on f (x, t) are as follows:
f_{1}(t) ≤ f (x, t) ≤ f_{2 }(t) uniformly in x ∈ Ω, where f_{l}, f_{2 }∈ C (ℝ) and we denote
In view of the condition , problem (1) is called generalized asymptotically linear at both zero and infinity. Clearly, u = 0 is a trivial solution of problem (1). It follows from and that the functional
is of C^{2 }on the space with the norm
where Under the condition , the critical points of I are solutions of problem (1). Let 0 < λ_{1 }< λ_{2 }< ··· < λ_{k }< ··· be the eigenvalues of and ϕ_{l}(x) > 0 be the eigenfunction corresponding to λ_{l}. In fact, . Let denote the eigenspace associated to λ_{k}. Throughout this article, we denoted by  · _{p }the L^{p }(Ω) norm.
If in the above condition is an eigenvalue of , then the problem (1) is called resonance at infinity. Otherwise, we call it nonresonance. A main tool of seeking the critical points of functional I is the mountain pass theorem (see [13]). To apply this theorem to the functional I in (2), usually we need the following condition [1], i.e., for some θ > 2 and M > 0,
It is well known that the condition (AR) plays an important role in verifying that the functional I has a "mountainpass" geometry and a related (PS)_{c }sequence is bounded in when one uses the mountain pass theorem.
If f(x, t) admits subcritical growth and satisfies (AR) condition by the standard argument of applying mountain pass theorem, we know that problem (1) has nontrivial solutions. Similarly, lase f(x, t) is of critical growth (see, for example, [47] and their references).
It follows from the condition (AR) that after a simple computation. That is, f(x, t) must be superlinear with respect to t at infinity. Noticing our condition , the nonlinear term f(x, t) is generalized asymptotically linear, not superlinear, with respect to t at infinity; which means that the usual condition (AR) cannot be assumed in our case. If the mountain pass theorem is used to seek the critical points of I, it is difficult to verify that the functional I has a "mountain pass" structure and the (PS)c sequence is bounded.
In [8], Zhou studied the following elliptic problem
where the conditions on f(x, t) are similar to and . He provided a valid method to verify the (PS) sequence of the variational functional, for above problem is bounded in (see also [9,10]).
To the authors' knowledge, there seems few results on problem (1) when f(x, t) is generalized asymptotically linear at infinity. However, the method in [8] cannot be applied directly to the biharmonic problems. For example, for the Laplacian problem, implies u, u_{+}, , where u_{+ }= max(u, 0), u_{ }= max(u, 0). We can use u_{+ }or u_{ }as a test function, which is helpful in proving a solution nonnegative. While for the biharmonic problems, this trick fails completely since does not imply u_{+}, (see [[11], Remark 2.1.10] and [12,13]). As far as this point is concerned, we will make use of the new methods to discuss in the following Lemma 2.2.
This fourthorder semilinear elliptic problem can be considered as an analogue of a class of secondorder problems which have been studied by many authors. In [14], there was a survey of results obtained in this direction. In [15], Micheletti and Pistoia showed that (P_{1}) admits at least two solutions by a variation of linking if f(x, u) is sublinear. Chipot [16] proved that the problem (P_{1}) has at least three solutions by a variational reduction method and a degree argument. In [17], Zhang and Li showed that (P_{1}) admits at least two nontrivial solutions by Morse theory and local linking if f(x, u) is superlinear and subcritical on u.
In this article, we consider multiple solutions of problem (1) in the nonresonance by using the mountain pass theorem and Morse theory. At first, we use the truncated skill and mountain pass theorem to obtain a positive solution and a negative solution of problem (1) under our more general conditions and with respect to the conditions (H_{1}) and (H_{3}) in [8]. In the course of proving existence of positive solution and negative solution, our conditions are general, but the proof of our compact condition is more simple than that in [8]. Furthermore, we can obtain a nontrivial solution when the nonlinear term f is nonresonance at the infinity by using Morse theory.
2 Main result and auxiliary lemmas
Let us now state the main result.
Theorem 2.1. Assume conditions and hold, and for some k ≥ 2, then problem (1) has at least three nontrivial solutions.
Consider the following problem
where
Lemma 2.2. I_{+ }satisfies the (PS) condition.
Proof. Let be a sequence such that as n → ∞. Note that
for all . Assume that u_{n}_{2 }is bounded, taking ϕ = u_{n }in (3). By , there exists c > 0 such that f_{+}(x, u_{n}(x)) ≤ cu_{n}(x), a.e. x ∈ Ω. So (u_{n}) is bounded in . If u_{n}_{2 }→ +∞, as n → ∞, set , then v_{n}_{2 }= 1. Taking ϕ = v_{n }in (3), it follows that v_{n} is bounded. Without loss of generality, we assume that v_{n }⇀ v in , then v_{n}→ v in L^{2}(Ω). Hence, v_{n }→ v a.e. in Ω. Dividing both sides of (3) by u_{n}_{2}, we get
Let ϵ > 0 and , and choose a constant M > 0 such that .
Let and ϕ ≥ 0. From (4), we have
Letting n → ∞, for all and ϕ ≥ 0 we have
Then we have
While let Δv = u, by the comparison maximum principle v ≥ 0. Since the definition of v_{n}, we have v ≢ 0 and we arrive at a contradiction by choosing ϕ = ϕ_{1 }in (5).
Since u_{n}_{2 }is bounded, from (3) we get the boundedness of u_{n}. A standard argument shows that {u_{n}} has a convergent subsequence in . Therefore, I_{+ }satisfies (PS) condition.
Lemma 2.3. Let ϕ_{1 }be the eigenfunction corresponding to λ_{1 }with ϕ_{1} = 1. If , then
(a) There exist ρ, β > 0 such that I_{+ }(u) ≥ β for all with u = ρ;
(b) I_{+}(tϕ_{1}) = ∞ as t → +∞.
Proof. By and , if , for any ε > 0, there exist A = A(ε) ≥ 0 and B = B(ε) such that for all (x, s) ∈ Ω × ℝ,
Choose ε > 0 such that . By (6), the Poincaré inequality and the Sobolev inequality, we get
So, part (a) holds if we choose u = ρ > 0 small enough.
On the other hand, if , take ε > 0 such that . By (7), we have
Since and ϕ_{l} = 1, it is easy to see that
and part (b) is proved.
Lemma 2.4. Let , where . If f satisfies , then
(i) the functional I is coercive on W, that is
and bounded from below on W,
(ii) the functional I is anticoercive on V.
Proof. For u ∈ W, by , for any ε > 0, there exists B_{l }= B_{l}(ε) such that for all (x, s) ∈ Ω × ℝ,
So we have
Choose ε > 0 such that . This proves (i).
(ii) When , it is easy to see that the conclusion holds.
Lemma 2.5. If , then I satisfies the (PS) condition.
Proof. Let be a sequence such that I (u_{n)} ≤ c, < I'(u_{n)}, ϕ > → 0.
Since
for all . If u_{n}_{2 }is bounded, we can take ϕ = u_{n}. By , there exists a constant c > 0 such that f (x, u_{n}(x)) ≤ cu_{n}(x), a.e. x ∈ Ω. So (u_{n}) is bounded in . If u_{n}_{2 }→ +∞, as n → ∞, set , then v_{n}_{2 }= 1. Taking ϕ = v_{n }in (9), it follows that vn is bounded. Without loss of generality, we assume vn ⇀ v in , then vn → v in L^{2}(Ω). Hence, vn → v a.e. in Ω. Dividing both sides of (9) by u_{n}_{2}, for any , we get
Then for a.e. x ∈ Ω and suitable ϵ, we have
as n → ∞. In fact, if v(x) ≠ 0, by , we have
and
as n → ∞. If v(x) = 0, we have
Since , by choosing ϕ = v in (10) and the Lebesgue dominated convergence theorem, we arrive at
From Fourier series theory, it is easy to see that v ≡ 0. It contradicts to .
It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the book [18] for more information on Morse theory.
Let H be a Hilbert space and I ∈ C^{l}(H, ℝ) be a functional satisfying the (PS) condition or (C) condition, and H_{q }(X, Y) be the qth singular relative homology group with integer coefficients. Let u_{0 }be an isolated critical point of I with I(u_{0}) = c, c ∈ ℝ, and U be a neighborhood of u_{0}. The group
is said to be the qth critical group of I at u_{0}, where I^{c }= {u ∈ H: I (u) ≤ c}.
Let K: = {u ∈ H: I'(u) = 0} be the set of critical points of I and a < inf I (K), the critical groups of I at infinity are formally defined by (see [19])
The following result comes from [18,19] and will be used to prove the result in this article.
Proposition 2.6 [19]. Assume that , I is bounded from below on and I(u) → ∞ as u → ∞ with . Then
3 Proof of the main result
Proof of Theorem 2.1. By Lemmas 2.2 and 2.3 and the mountain pass theorem, the functional I_{+ }has a critical point u_{l }satisfying I_{+}(u_{l}) ≥ β. Since I_{+ }(0) = 0, u_{l }≠ 0 and by the maximum principle, we get u_{l }> 0. Hence u_{l }is a positive solution of the problem (1) and satisfies
Using the results in [18], we obtain
Similarly, we can obtain another negative critical point u_{2 }of I satisfying
Since , the zero function is a local minimizer of I, then
On the other hand, by Lemmas 2.4, 2.5 and the Proposition 2.6, we have
Hence I has a critical point u_{3 }satisfying
Since k ≥ 2, it follows from (13)(17) that u_{l}, u_{2}, and u_{3 }are three different nontrivial solutions of the problem (1).
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author would like to thank the referees for valuable comments and suggestions in improving this article. This study was supported by the National NSF (Grant No. 10671156) of China.
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