We consider the nonlinear biharmonic equation with variable coefficient and polynomial growth nonlinearity and Dirichlet boundary condition. We get two theorems. One theorem says that there exists at least one bounded solution under some condition. The other one says that there exist at least two solutions, one of which is a bounded solution and the other of which has a large norm under some condition. We obtain this result by the variational method, generalized mountain pass geometry and the critical point theory of the associated functional.
MSC: 35J20, 35J25, 35Q72.
Keywords:biharmonic boundary value problem; polynomial growth; variational method; generalized mountain pass geometry; critical point theory; condition
Let Ω be a bounded domain in with smooth boundary ∂Ω and be a square integrable function space defined on Ω. Let Δ be the elliptic operator and be the biharmonic operator. Let . In this paper we study the following nonlinear biharmonic equation with Dirichlet boundary condition:
We assume that g satisfies the following conditions:
Remark 1.1 The real number ξ in the definition (g3) is not automatically nonnegative. The reason is as follows.
Remark 1.3 (i) Assumption (g4) implies that (1.1) has a trivial solution.
The eigenvalue problem
has infinitely many eigenvalues , , and corresponding eigenfunctions , , suitably normalized with respect to the inner product, where each eigenvalue is repeated as often as its multiplicity. The eigenvalue problem
has at least two nontrivial solutions when (, and ) or (, and ). The authors obtained these results by using the variational reduction method. The authors  also proved that when , and , (1.3) has at least three nontrivial solutions by using degree theory. Tarantello  also studied
She showed that if and , then (1.4) has a negative solution. She obtained this result by degree theory. Micheletti and Pistoia  also proved that if and then (1.4) has at least four solutions by the variational linking theorem and Leray-Schauder degree theory.
In this paper we are trying to find the weak solutions of (1.1), that is,
where the space H is introduced in Section 2. Let us set
Theorem 1.2Assume that, gsatisfies (g1)-(g4), is not bounded and there exists a smallsuch that. Then (1.1) has at least two solutions, (i) one of which is nontrivial and bounded, and (ii) the other of which has a large norm such that
The outline of Theorem 1.1 and Theorem 1.2 is as follows: In Section 2, we prove that the corresponding functional of (1.1), which is introduced in (2.1), is continuous and Fréchet differentiable and satisfies the condition. In Section 3, we prove Theorem 1.1. In Section 4, we prove Theorem 1.2 by the variational method, the generalized mountain pass geometry and the critical point theory.
2 Palais-Smale condition
Then this is a Banach space with a norm
which are proved in .
We are looking for the weak solutions of (1.1). The weak solutions of (1.1) coincide with the critical points of the associated functional
If we set
The proof of Proposition 2.1 is the same as that of Appendix B in .
Proposition 2.2 (Palais-Smale condition)
Thus, by (2.3), we have
3 Proof of Theorem 1.1
We recall the generalized mountain pass geometry.
We have the following generalized mountain pass geometrical assumptions.
Thus by (g2), (g4), and the Hölder inequality, we have
Thus we have
Proof of Theorem 1.1 By Proposition 2.1 and Proposition 2.2, and satisfies the Palais-Smale condition. By Lemma 3.1, there are constants , and a bounded neighborhood of 0 such that , and there is an and such that if , then , and there exists such that and . By the generalized mountain pass theorem, has a critical value . Moreover, b can be characterized as
In fact, we have
We can write
4 Proof of Theorem 1.2
Assume that is not bounded and there exists an such that . By Proposition 2.1 and Proposition 2.2, and satisfies the Palais-Smale condition. By Lemma 3.1 and the generalized mountain pass theorem, has a critical value b with critical point such that . If is sufficiently small, by (3.1), we have
Let us set
Proof of Theorem 1.2(ii) We assume that is not bounded and there exists an such that . By Proposition 2.1 and Proposition 2.2, and satisfies the Palais-Smale condition. By Lemma 4.1, there exists an such that for . We note that . By Lemma 4.1 and the generalized mountain pass theorem, for n large enough is a critical value of I and . Let be a critical point of I such that . Then for each real number M, . In fact, by contradiction, and imply that
The authors declare that they have no competing interests.
TJ and Q-HC participated in the sequence alignment and drafted the manuscripted. Both authors read and approved the final manuscript.
This work (Choi) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2013010343).
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