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:
where is a continuous function which changes sign in Ω.
We assume that g satisfies the following conditions:
(g2) there are constants such that
where if ,
(g3) there exists a constant such that
(g4) as .
We note that (g3) implies the existence of the positive constants , , such that
Remark 1.1 The real number ξ in the definition (g3) is not automatically nonnegative. The reason is as follows.
Since and , and . By , we have two cases: one case is that and . The other case is that and . Thus ξ is not nonnegative.
Remark 1.2 We obtain the boundedness of as follows.
By the condition (g3), for . Since , , and in (1.2),
Thus we obtain the boundedness of .
Remark 1.3 (i) Assumption (g4) implies that (1.1) has a trivial solution.
(ii) If , (g2) can be dropped. If , it suffices that
where as .
(iii) If and , where and is a small number, then (g1)-(g4) are satisfied.
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 also infinitely many eigenvalues , and corresponding eigenfunctions , . We note that , and that for .
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
Since changes sign, the open subsets and are nonempty. Now we can write . Our main results are as follows.
Theorem 1.1Assume that , gsatisfies (g1)-(g4) and is bounded. Then (1.1) has at least one bounded nontrivial solution.
Theorem 1.2Assume that , gsatisfies (g1)-(g4), is not bounded and there exists a small such 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
Any element u in can be written as
We define a subspace H of as follows:
Then this is a Banach space with a norm
Since and c is fixed, we have
(i) implies ,
(ii) , for some ,
(iii) if and only if ,
which are proved in .
Then . Let be the orthogonal projection on and be the orthogonal projection on .
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
By (g1) and (g2), I is well defined. By Proposition 2.1, and I is Fréchet differentiable in H.
Proposition 2.1Assume that , , and thatgsatisfies (g1)-(g4). Then is continuous and Fréchet differentiable inHwith Fréchet derivative
If we set
then is continuous with respect to weak convergence, is compact, and
This implies that and is weakly continuous.
The proof of Proposition 2.1 is the same as that of Appendix B in .
Proposition 2.2 (Palais-Smale condition)
Assume that , , andgsatisfies (g1)-(g4). We also assume that is bounded or that there exists an such that . Then satisfies the Palais-Smale condition.
Proof Suppose that is a sequence with and as . Then by (g2), (g3), and the Hölder inequality and the Sobolev Embedding Theorem, for large m and with , we have
Since is bounded or there exists an such that ; we have
for large m and all , choosing gives
Taking in (2.4) yields
Thus, by (2.3), we have
from which the boundedness of follows. Thus converges weakly in H. Since with compact and the weak convergence of imply the strong convergence of and hence condition holds. □
3 Proof of Theorem 1.1
We shall show that satisfies the generalized mountain pass geometrical assumptions.
We recall the generalized mountain pass geometry.
Let , where and is finite dimensional. Suppose that , satisfies the Palais-Smale condition, and
(i) there are constants and a bounded neighborhood of 0 such that , and
(ii) there is an and such that if , then .
Then I possesses a critical value . Moreover b can be characterized as
Let . Then is a subspace of H such that
We have the following generalized mountain pass geometrical assumptions.
Lemma 3.1Assume that andgsatisfies (g1)-(g4). Then
(i) there are constants , and a bounded neighborhood of 0 such that , and
(ii) there is an and such that if , then , and
(iii) there exists such that and .
Proof (i) Let . Then
Thus by (g2), (g4), and the Hölder inequality, we have
for . Since , there exist and such that if , then .
(ii) Let be a ball with radius , e be a fixed element in and . Then , , . We note that
Thus we have
Since , there exists such that if , then .
(iii) If we choose such that , in Ω and , then we have
for all . Since , for great enough, is such that and . □
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
We denote by a critical point of I such that . We claim that there exists a constant such that
In fact, we have
Since , b is bounded: .
We can write
where . Thus we have
for some constants , from which we conclude that is bounded and the proof of Theorem 1.1 is complete. □
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
for , from which we can conclude that is bounded and the proof of Theorem 1.2(i) is complete.
Next we shall prove Theorem 1.2(ii). We may assume that for all . Let us set , .
Lemma 4.1Assume thatgsatisfies (g1)-(g4), is not bounded and there exists an such that . Then there exists an such that
Proof Let us choose such that , in Ω and . Then, by (g2), (g4), and the Hölder inequality, we have
for small . Since , there exist great enough for each n and an such that and if and , so the lemma is proved. □
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
which means that is bounded. This is absurd because of the fact that . Thus we complete the proof. □
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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