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; condition
Let D be an open subset in with compact complement , . In this paper we investigate the multiplicity of the solutions for a class of the system of nonlinear wave equations with the Dirichlet boundary condition and periodic condition:
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
This is a complete normed space with a norm
Since is unbounded from above and from below and has no finite accumulation point, it is convenient for the following to rearrange the eigenvalues by increasing magnitude: from now on we denote by the sequence of negative eigenvalues of (2.1), by the sequence of positive ones, so that
We will denote by the sequence all the sequences and . Let be an orthonormal system of the eigenfunctions associated with the eigenvalues . We will denote by the sequence the sequences , . Let be the span of closure of eigenfunctions associated with positive eigenvalues and be the span of closure of eigenfunctions associated with negative eigenvalues. Let H be the n Cartesian product space of E, i.e.,
is positive definite and negative definite, respectively. Then
Let us introduce an open set of the Hilbert space H as follows:
Let us consider the functional on X
Thus we have
Thus by (2.3), we have
Proof To prove the conclusion, it suffices to prove that
By (G1) and (G2), there exists a constant B such that
Thus we have
Thus we have
Thus by Fatou’s lemma, we have
so we prove the lemma. □
We recall the critical point theorem for the indefinite functional (cf.).
Theorem 2.1 (Critical point theorem for the indefinite functional)
3 Proof of Theorem 1.1
Lemma 3.1 (Palais-Smale condition)
where and is a compact operator. We claim that the sequence , up to a subsequence, converges. It suffices to prove that the sequence is bounded in X. By contradiction, we suppose that . Then, for large k, we have
It follows from (3.2) that
Thus we have
and by (3.5), W is the weak solution of the equation
We note that
Thus . Thus , which is absurd to the fact that . Thus is bounded. Thus the subsequence, up to a subsequence, converges weakly to U in X. By Lemma 2.2, and that is bounded. Since is compact and (3.1) holds, converges strongly to U. Thus we prove the lemma. □
Proof of Theorem 1.1 By Lemma 2.1, is continuous and Fréchet differentiable in X and, moreover, . By Lemma 2.2, if and weakly in X with , then . By Lemma 3.1, satisfies the (PS) condition. By Lemma 3.2, there exist sets with radius , and constant such that , Q is bounded and , and and ∂Q link. By the critical point theorem, possesses a critical value . Thus (1.1) has at least one nontrivial weak solution. Thus we prove Theorem 1.1 □
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the final manuscript.
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 (KRF-2010-0023985).
Benci, V, Rabinowitz, PH: Critical point theorems for indefinite functionals. Invent. Math.. 52, 241–273 (1979). Publisher Full Text