The eigenvalue problem
(2.1)
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
(2.2)
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 2.1 Assume that G satisfies conditions (G1)-(G2). Then is continuous and Fréchet differentiable in X with Fréchet derivative
Moreover, . That is, .
Proof First we prove that is continuous. For ,
We have
(2.3)
Thus we have
Next we shall prove that is Fréchet differentiable in X. For ,
Thus by (2.3), we have
(2.4)
Similarly, it is easily checked that . □
Let
Lemma 2.2 Assume that G satisfies conditions (G1)-(G2). Let and weakly in X with . 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
Hence
Since the embedding is compact, we have
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)
Let X be 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)
Q is bounded and ,
-
(iii)
S and ∂Q link.
Then I possesses a critical value .