Abstract
In this paper, the existence of at least one nontrivial solution for a class of fourthorder elliptic equations with the Navier boundary value conditions is established by using the linking methods.
Keywords:
biharmonic; Navier boundary value problems; local linking1 Introduction
Consider the following Navier boundary value problem:
where
The conditions imposed on
(H_{1}) , and there are constants
where
(H_{2})
(H_{3})
(H_{4}) There is a constant
where
(H_{5}) For some
or
This fourthorder semilinear elliptic problem has been studied by many authors. In [1], there was a survey of results obtained in this direction. In [2], Micheletti and Pistoia showed that (1.1) admits at least two solutions by a variation
of linking if
We will use linking methods to give the existence of at least one nontrivial solution for (1.1).
Let X be a Banach space with a direct sum decomposition
The function
It is clear that 0 is a critical point of I.
The notion of local linking generalizes the notions of local minimum and local maximum.
When 0 is a nondegenerate critical point of a functional of class
The condition of local linking was introduced in [8] under stronger assumptions
Under those assumptions, the existence of nontrivial critical points was proved for functionals which are
(a) bounded below [8],
(b) superquadratic [8] and
(c) asymptotically quadratic [9].
The cases (a), (b) and (c) were considered in [10] with only conditions (1.2) and (1.3), and three theorems about critical points were proved. Using these theorems, the author in [10] proved the existence of at least one nontrivial solution for the secondorder elliptic boundary value problem with the Dirichlet boundary value condition. In the present paper, we also use the three theorems in [10] and the linking technique to give the existence of at least one nontrivial solution for the biharmonic problem (1.1) under a weaker condition. The main results are as follows.
Theorem 1.1Assume the conditions (H_{1})(H_{4}) hold. Iflis an eigenvalue of −△ (with the Dirichlet boundary condition), assume also (H_{5}). Then the problem (1.1) has at least one nontrivial solution.
We also consider asymptotically quadratic functions.
Let
(H_{6})
Theorem 1.2Assume the conditions (H_{1}), (H_{6}) and one of the following conditions:
(A_{1})
(A_{2})
(A_{3})
Then the problem (1.1) has at least one nontrivial solution.
2 Preliminaries
Let X be a Banach space with a direct sum decomposition
Consider two sequences of a subspace:
such that
For every multiindex
A sequence
Definition 2.1 Let I be locally Lipschitz on X and
contains a subsequence which converges to a critical point of I.
Definition 2.2 Let I be locally Lipschitz on X and
contains a subsequence which converges to a critical point of I.
Remark 2.1
1. The
2. When the
3. Without loss of generality, we assume that the norm in X satisfies
Definition 2.3 Let X be a Banach space with a direct sum decomposition
The function
Lemma 2.1 (see [10])
Suppose that
(B_{1}) Ihas a local linking at 0 and
(B_{2}) Isatisfies
(B_{3}) Imaps bounded sets into bounded sets;
(B_{4}) for every
Remark 2.2 Assume I satisfies the
Let X be a real Hilbert space and let
where A is a bounded selfadjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.
We assume that there exist an orthogonal decomposition,
and two sequences of finitedimensional subspaces,
such that
For every multiindex
by
Lemma 2.2 (see [10])
Isatisfies the following assumptions:
(i) Ihas a local linking at 0 with respect to
(ii) there exists a compact selfadjoint operator
(iii)
(iv) for infinitely many multipleindices
ThenIhas at least two critical points.
3 The proof of main results
Proof of Theorem 1.1 (1) We shall apply Lemma 2.1 to the functional
defined on
Then other case is similar and simple.
Let
By the condition (H_{1}) and Sobolev inequalities, it is easy to see that the functional I belongs to
(2) We claim that I has a local linking at 0 with respect to
It follows from (H_{1}) and (H_{2}) that for any
Hence, we obtain
where
Let
From (3.2), we have
for all
On the other hand, we have
for all
for all
where
for all
for
(3) We claim that I satisfies
and
Let
If
For any
So, for n large enough,
where ϵ is a small enough constant.
That is,
Therefore, using (H_{3}), we have
This contradicts (3.5).
If
From (3.4), we obtain
By (3.8), the righthand side of (3.9)
In any case, we obtain a contradiction. Therefore,
Finally, we claim that for every
By (H_{2}) and (H_{3}), there exists large enough M such that
So, for any
Hence, our claim holds. □
Proof of Theorem 1.2 We omit the proof which depends on Lemma 2.2 and is similar to the preceding one. □
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
The author would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.
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