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:
where . Let . We assume that G satisfies the following conditions:
(G1) There exists such that
(G2) There is a neighborhood Z of C in such that
where is the distance function from U to C and is a constant. The system (1.1) can be rewritten as
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 , , given by
Let Ω be the square , and be the Hilbert space defined by
The set of functions is an orthonormal basis in . Let us denote an element v, in , as
and we define a subspace E of as
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.,
Let and be the subspaces on which the functional
is positive definite and negative definite, respectively. Then
Let be the projection from H onto and be the projection from H onto . The norm in H is given by
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.
In this paper we are trying to find the weak solutions of system (1.1), that is, such that
for all , i.e.,
Let us introduce an open set of the Hilbert space H as follows:
Let us consider the functional on X
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 .
Lemma 2.1Assume thatGsatisfies conditions (G1)-(G2). Then is continuous and Fréchet differentiable inXwith Fréchet derivative
Moreover, . That is, .
Proof First we prove that is continuous. For ,
Thus we have
Next we shall prove that is Fréchet differentiable in X. For ,
Thus by (2.3), we have
Similarly, it is easily checked that . □
Lemma 2.2Assume thatGsatisfies conditions (G1)-(G2). Let and weakly inXwith . Then .
Proof To prove the conclusion, it suffices to prove that
Since is bounded from below, it suffices to prove that there is a subset of Ω such that
means that there exists such that . Let us set
By (G1) and (G2), there exists a constant B such that
Thus we have
for all . By Schwarz’s inequality, we have
Thus we have
Since the embedding is compact, 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)
LetXbe a real Hilbert space with and . Suppose that satisfies (PS), and
(I1) , where and is bounded and self-adjoint, ,
(I2) is compact, and
(I3) there exists a subspace and sets , and constants such that
(i) and ,
(ii) Qis bounded and ,
ThenIpossesses 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 thatGsatisfies conditions (G1) and (G2). Then satisfies the (PS) condition inX.
Proof We shall prove the lemma by contradiction. We suppose that there exists a sequence satisfying and
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
Let us set . Then , and hence the subsequence , up to a subsequence, converges weakly to W with . By (3.1), we have
Letting in (3.4), by (3.3), we have
Thus we have
and by (3.5), W is the weak solution of the equation
We claim that , where and is the kernel of A. In fact, let , . Then
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. □
Lemma 3.2Assume thatGsatisfies conditions (G1) and (G2). Then there exist sets with radius , and constants such that
(i) and ,
(ii) Qis bounded and ,
Proof (i) Let us choose . Then . By (G1), is bounded above and there exists a constant
for . Then there exist constants and such that if , then .
(ii) Let us choose . Let . Then , , . We note that:
By (G2), is bounded from below. Thus by Lemma 2.2, there exists a constant such that if , then we have
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 □
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