Weak solutions for the singular potential wave system
© Jung and Choi; licensee Springer. 2014
Received: 13 February 2013
Accepted: 9 December 2013
Published: 7 January 2014
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.
(G1) There exists such that
(G2) There is a neighborhood Z of C in such thatwhere is the distance function from U to C and is a constant. The system (1.1) can be rewritten as(1.2)
Our main result is the following.
Theorem 1.1 Assume that the nonlinear term G satisfies 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
where , , .
Let be a sequence of closed finite dimensional subspace of H with the following assumptions: , where , for all n ( and are subspaces of H), , , is dense in H.
where and . The Euler equation for (2.1) is (1.1). By the following Lemma 2.1, , and so the weak solutions of system (1.1) coincide with the critical points of the associated functional .
Moreover, . That is, .
Similarly, it is easily checked that . □
Lemma 2.2 Assume that G satisfies conditions (G1)-(G2). Let and weakly in X with . Then .
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)
(I1) , where and is bounded and self-adjoint, ,
(I2) is compact, and
(I3) there exists a subspace and sets , and constants such that
Q is bounded and ,
S and ∂Q link.
Then I possesses a critical value .
3 Proof of Theorem 1.1
We shall show that the functional satisfies the geometric assumptions of the critical point theorem for indefinite functional.
Lemma 3.1 (Palais-Smale condition)
Assume that G satisfies conditions (G1) and (G2). Then satisfies the (PS) condition in X.
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. □
Q is bounded and ,
and ∂Q link.
- (ii)Let us choose . Let . Then , , . We note that:
We can choose a constant such that if , then . 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 □
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).
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