Abstract
We investigate the existence of weak solutions for a class of the system of wave equations with singular potential nonlinearity. We obtain a theorem which shows the existence of nontrivial weak solution for a class of the wave system with singular potential nonlinearity and the Dirichlet boundary condition. We obtain this result by using the variational method and critical point theory for indefinite functional.
MSC: 35L51, 35L70.
Keywords:
class of wave system; singular potential nonlinearity; Dirichlet boundary condition; variational method; critical point theorem for indefinite functional;1 Introduction
Let D be an open subset in
where
(G1) There exists
(G2) There is a neighborhood Z of C in
where
where
Remark We have a simple example satisfying the above conditions (G1)(G2):
Our main result is the following.
Theorem 1.1Assume that the nonlinear termGsatisfies conditions (G1)(G2). Then system (1.1) has at least one nontrivial weak solution.
For the proof of Theorem 1.1, we approach the variational method and use the critical point theory for indefinite functional. In Section 2, we introduce a Banach space and the associated functional I of (1.1), and recall the critical point theory for indefinite functional. In Section 3, we prove that I satisfies the geometric assumptions of the critical point theorem for indefinite functional and prove Theorem 1.1.
2 Variational approach
The eigenvalue problem
has infinitely many eigenvalues
and corresponding normalized eigenfunctions
Let Ω be the square
The set of functions
and we define a subspace E of
This is a complete normed space with a norm
Since
We will denote by the sequence
Let
is positive definite and negative definite, respectively. Then
Let
where
Let
In this paper we are trying to find the weak solutions
for all
Let us introduce an open set of the Hilbert space H as follows:
Let us consider the functional on X
where
Lemma 2.1Assume thatGsatisfies conditions (G1)(G2). Then
Moreover,
Proof First we prove that
We have
Thus we have
Next we shall prove that
Thus by (2.3), we have
Similarly, it is easily checked that
Let
Lemma 2.2Assume thatGsatisfies conditions (G1)(G2). Let
Proof To prove the conclusion, it suffices to prove that
Since
By (G1) and (G2), there exists a constant B such that
Thus we have
for all
Thus we have
Hence
Since the embedding
Thus by Fatou’s lemma, we have
Thus
Thus
so we prove the lemma. □
We recall the critical point theorem for the indefinite functional (cf.[1]).
Let
Theorem 2.1 (Critical point theorem for the indefinite functional)
LetXbe a real Hilbert space with
(I1)
(I2)
(I3) there exists a subspace
(i)
(ii) Qis bounded and
(iii) Sand∂Qlink.
ThenIpossesses a critical value
3 Proof of Theorem 1.1
We shall show that the functional
Lemma 3.1 (PalaisSmale condition)
Assume thatGsatisfies conditions (G1) and (G2). Then
Proof We shall prove the lemma by contradiction. We suppose that there exists a sequence
or equivalently
where
It follows from (3.2) that
Let us set
Letting
Thus we have
Thus
and by (3.5), W is the weak solution of the equation
We claim that
We note that
Thus
Let
Lemma 3.2Assume thatGsatisfies conditions (G1) and (G2). Then there exist sets
(i)
(ii) Qis bounded and
(iii)
Proof (i) Let us choose
for
(ii) Let us choose
By (G2),
We can choose a constant
Proof of Theorem 1.1 By Lemma 2.1,
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the final manuscript.
Acknowledgements
This work (Tacksun Jung) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF20100023985).
References

Benci, V, Rabinowitz, PH: Critical point theorems for indefinite functionals. Invent. Math.. 52, 241–273 (1979). Publisher Full Text