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Fourth order elliptic boundary value problem with nonlinear term decaying at the origin
Journal of Inequalities and Applications volume 2013, Article number: 432 (2013)
We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.
Let Ω be a bounded domain in with smooth boundary ∂ Ω. Let and be a function. In this paper, we consider the number of the weak solutions for the following fourth order elliptic problem with the Dirichlet boundary condition
We assume that satisfies the following:
(g2) , uniformly with respect to ,
(g3) there exists such that .
The eigenvalue problem
has infinitely many eigenvalues , , which is repeated as often as its multiplicity, and the corresponding eigenfunctions , suitably normalized with respect to inner product. The eigenvalue problem
has also infinitely many eigenvalues , and corresponding eigenfunctions , . We note that
Furthermore, we assume that satisfies .
Jung and Choi  proved that (1.1) has at least one nontrivial solution, when and g satisfies the condition (g1), (g2) and additional conditions
(g3)′ there exists such that ,
(g4)′ there exist a constant and an element such that , and ,
by reducing problem (1.1) to the problem with bounded nonlinear term and then applying the maximum principle for the elliptic operator −Δ and two times and the mountain pass theorem in the critical point theory. Jung and Choi  showed the existence of at least two solutions, one of which is a bounded solution and a large norm solution of (1.1), when is polynomial growth or exponential growth nonlinear term. The authors proved these results by the variational method and the mountain pass theorem. For the constant coefficient semilinear case Choi and Jung  showed that the problem
has at least two nontrivial solutions, when , and or when , and . The authors obtained these results by using the variational reduction method. The authors  also proved that when , and , (1.2) has at least three nontrivial solutions by using the degree theory. Tarantello  also studied the problem
She showed that if and , then (1.3) has a negative solution. She obtained this result by the degree theory. Micheletti and Pistoia  also proved that if and , then (1.3) has at least three solutions by the variational linking theorem and Leray-Schauder degree theory.
In this paper, we are trying to find weak solutions of (1.1), that is,
where H is introduced in Section 2.
We consider the associated functional of (1.1)
where . By (g1), I is well defined.
Our main result is the following.
Theorem 1.1Assume that, , andgsatisfies the conditions (g1)-(g3). Then (1.1) has at least one bounded weak solution.
We prove Theorem 1.1 by approaching the variational method and using the mountain pass theorem for the reduced fourth order elliptic problem with bounded nonlinear term. The outline of the proof of Theorem 1.1 is as follows: In Section 2, we prove that functional and the functional I satisfies the Palais-Smale condition. In Section 3, we show that the functional I satisfies the generalized mountain pass theorem, and so, prove that I has at least one nontrival critical point, from which we prove Theorem 1.1.
2 Variational approach
Let be a square integrable function space defined on Ω. Any element u in can be written as
We define a subspace H of as follows
Then this is a complete normed space with a norm
Since and c is fixed, we have and
for some ,
if and only if ,
which is proved in .
Then , for , . Let be the orthogonal projection from H onto and be the orthogonal projection from H onto . We can write , , for .
By the following Lemma 2.1, the weak solutions of (1.1) coincide with the critical points of the associated functional .
Lemma 2.1Assume that, , andgsatisfies the conditions (g1)-(g3). Thenis continuous, and Fréchet differentiable inHwith Fréchet derivative
If we set
thenis continuous with respect to weak convergence, is compact and
this implies thatandis weakly continuous.
The proof of Lemma 2.1 has the similar process to that of the proof in Appendix B in .
Now, we shall show that satisfies the Palais-Smale condition.
Lemma 2.2Assume that, , andgsatisfies the conditions (g1)-(g3). Then the functionalIsatisfies the Palais-Smale condition: Any sequenceinH, for whichandas, possesses a convergent subsequence.
Proof Let us choose . By and (g1), is bounded. Then we have
Since u is bounded and is bounded, is bounded from below. Thus, I satisfies the condition. □
3 Proof of Theorem 1.1
Now, we recall the generalized mountain pass theorem (cf. Theorem 5.3 in ).
Theorem 3.1 (Generalized mountain pass theorem)
LetHbe a real Banach space with, whereand is finite-dimensional. Suppose thatsatisfiescondition, and
there are constantsand a bounded neighborhoodof 0 such that, and
there is anandsuch that if, then.
ThenIpossesses a critical value. Moreover, bcan be characterized as
We shall show that the functionalIsatisfies the generalized mountain pass geometrical assumptions.
Let. Thenis a subspace ofHsuch that
Lemma 3.1Assume thatandgsatisfies (g1)-(g3). Then
there are constantsand a bounded neighborhoodof 0 such that, and
there is anandsuch that if, then, and
there existssuch thatand.
Proof (i) Let . We note that
Since is bounded, there exists a constant such that . Thus, we have
for . There exist and such that if , then .
Let us choose an element . Let . Then , , . We note that
Thus, we have
for . Then there exists such that if , then , from which we can choose an element such that .
If we choose , then by (ii), . □
Proof of Theorem 1.1 We will show that has a nontrivial critical point by the generalized mountain pass theorem. By Lemma 2.1, is continuous and Fréchet differentiable in H. By Lemma 2.2, the functional I satisfies condition. We note that . By Lemma 3.1, there are constants and a bounded neighborhood of 0 such that , and there is an and such that if . Let us set
By the generalized mountain pass theorem, I possesses a critical value . Moreover, b can be characterized as
Thus, we prove that I has at least one nontrivial critical point. We denote by a critical point of I such that . We claim that b is bounded. In fact, by (iii) of Lemma 3.1, we have
and by (g3),
for some constant . Since , b is bounded:
We claim that is bounded. In fact, by contradiction, and for any , imply that
is not bounded, which is absurd to the fact that is bounded. Thus, is bounded, so (1.1) has at least one bounded weak solution. Thus, we prove Theorem 1.1. □
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This work (Tacksun Jung) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2010-0023985).
The authors declare that they have no competing interests.
TJ carried out the studies for the existence of weak solutions of the fourth order elliptic boundary value problem, participated in the sequence alignment and drafted the manuscript. QC participated in the sequence alignment and drafted the manuscript.
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Jung, T., Choi, QH. Fourth order elliptic boundary value problem with nonlinear term decaying at the origin. J Inequal Appl 2013, 432 (2013). https://doi.org/10.1186/1029-242X-2013-432
- fourth order elliptic boundary value problem
- nonlinear term decaying at the origin
- bounded nonlinear term
- variational method
- generalized mountain pass theorem