Abstract
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;1 Introduction
Let Ω be a bounded domain in
where
We assume that g satisfies the following conditions:
(g1)
(g2) there are constants
where
(g3) there exists a constant
(g4)
We note that (g3) implies the existence of the positive constants
Remark 1.1 The real number ξ in the definition (g3) is not automatically nonnegative. The reason is as follows.
Since
Remark 1.2 We obtain the boundedness of
By the condition (g3),
Thus we obtain the boundedness of
Remark 1.3 (i) Assumption (g4) implies that (1.1) has a trivial solution.
(ii) If
where
(iii) If
The eigenvalue problem
has infinitely many eigenvalues
has also infinitely many eigenvalues
Khanfir and Lassoued [1] showed the existence of at least one solution for the nonlinear elliptic boundary
problem when g is locally Hölder continuous on
has at least two nontrivial solutions when (
She showed that if
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
and let
Since
Theorem 1.1Assume that
Theorem 1.2Assume that
The outline of Theorem 1.1 and Theorem 1.2 is as follows: In Section 2, we prove that
the corresponding functional
2 PalaisSmale condition
Any element u in
We define a subspace H of
Then this is a Banach space with a norm
Since
(i)
(ii)
(iii)
which are proved in [6].
Let
Then
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,
Proposition 2.1Assume that
If we set
then
This implies that
The proof of Proposition 2.1 is the same as that of Appendix B in [7].
Proposition 2.2 (PalaisSmale condition)
Assume that
Proof Suppose that
Since
Moreover since
for large m and all
Taking
Thus, by (2.3), we have
from which the boundedness of
3 Proof of Theorem 1.1
We shall show that
We recall the generalized mountain pass geometry.
Let
(i) there are constants
(ii) there is an
Then I possesses a critical value
where
Let
Let
We have the following generalized mountain pass geometrical assumptions.
Lemma 3.1Assume that
(i) there are constants
(ii) there is an
(iii) there exists
Proof (i) Let
Thus by (g2), (g4), and the Hölder inequality, we have
for
(ii) Let
Thus we have
Since
(iii) If we choose
for all
Proof of Theorem 1.1 By Proposition 2.1 and Proposition 2.2,
where
We denote by
where
In fact, we have
and
Since
We can write
where
for some constants
4 Proof of Theorem 1.2
Assume that
for
Next we shall prove Theorem 1.2(ii). We may assume that
Lemma 4.1Assume thatgsatisfies (g1)(g4),
Proof Let us choose
for small
Let us set
and
Proof of Theorem 1.2(ii) We assume that
which means that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
TJ and QHC participated in the sequence alignment and drafted the manuscripted. Both authors read and approved the final manuscript.
Acknowledgements
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 (KRF2013010343).
References

Khanfir, S, Lassoued, L: On the existence of positive solutions of a semilinear elliptic equation with change of sign. Nonlinear Anal., Theory Methods Appl.. 22(11), 1309–1314 (1994). Publisher Full Text

Choi, QH, Jung, T: Multiplicity results on nonlinear biharmonic operator. Rocky Mt. J. Math.. 29(1), 141–164 (1999). Publisher Full Text

Jung, TS, Choi, QH: Multiplicity results on a nonlinear biharmonic equation. Nonlinear Anal., Theory Methods Appl.. 30(8), 5083–5092 (1997). Publisher Full Text

Tarantello, G: A note on a semilinear elliptic problem. Differ. Integral Equ.. 5(3), 561–565 (1992)

Micheletti, AM, Pistoia, A: Multiplicity results for a fourthorder semilinear elliptic problem. Nonlinear Anal. TMA. 31(7), 895–908 (1998). Publisher Full Text

Choi, QH, Jung, T: Multiplicity of solutions and source terms in a fourth order nonlinear elliptic equation. Acta Math. Sci.. 19(4), 361–374 (1999)

Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)