- Research Article
- Open Access
Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems
© Kuan-Ju Chen. 2009
- Received: 16 December 2008
- Accepted: 6 July 2009
- Published: 17 August 2009
We proved a multiplicity result for strongly indefinite semilinear elliptic systems in , in where and are positive numbers which are in the range we shall specify later.
- Bilinear Form
- Invariant Subspace
- Bounded Linear Operator
- Multiple Solution
- Convergent Subsequence
has to be redefined, and we then need fractional Sobolev spaces.
Hence the energy functional is strongly indefinite, and we shall use the generalized critical point theorem of Benci  in a version due to Heinz  to find critical points of . And there is a lack of compactness due to the fact that we are working in .
We shall propose herein a result similar to  for problem (1.1).
It is known that is an isomorphism, and so we denote by the inverse of .
We can then prove that has two eigenvalues and , whose corresponding eigenspaces are
In the sequel denotes the norm in , and we denote by the weighted function spaces with the norm defined on by . According to the properties of interpolation space, we have the following embedding theorem.
Then the inclusion of into is compact if .
where ; hence is well defined.
so that by H lder's inequality, we observe that, for any , we can choose a so that the integral over ( ) is smaller than for all , while for this fixed , by strong convergence of to in on any bounded region, the integral over ( ) is smaller than for large enough. We thus have proved that strongly in ; that is, the inclusion of into is compact if .
and we assume that
so that, under assumption (H), Theorem 2.1 holds, respectively, with and , and and ; that is, the inclusion of into and the inclusion of into are compact.
denote the energy of . It is well known that under assumption (H) the energy functional is well defined and continuously differentiable on , and for all we have
and it is also well known that the critical points of are weak solutions of problem (3.1). The main theorem is the following.
Under assumption (H), problem (3.1) possesses infinitely many solutions .
Since the functional are strongly indefinite, a modified multiplicity critical points theorem Heinz  which is the generalized critical point theorem of Benci  will be used. For completeness, we state the result from here.
(see ) Let be a real Hilbert space, and let be a functional with the following properties:
where is an invertible bounded self-adjoint linear operator in and where is such that and the gradient is a compact operator;
(ii) is even, that is ;
suppose that there is an -dimensional linear subspace of ( ) such that for the spaces , one has
(iv) such that , ;
a similar result holds when , and one takes , .
The functional defined in (3.6) satisfies conditions (ii), (iv), and (v) of Theorem 3.2.
and since , , we conclude that for with small.
The same arguments can be applied if . So the result follows from (3.16).
A sequence is said to be the Palais-Smale sequence for (PS)-sequence for short) if uniformly in and in . We say that satisfies the Palais-Smale condition (PS)-condition for short) if every (PS)-sequence of is relatively compact in .
Under assumption (H), the functional satisfies the (PS)-condition.
Since and , we conclude that both and are bounded, and consequently and are also bounded in terms of (3.24).
and by Theorem 2.1, we conclude that strongly in and strongly in .
Proof of Theorem 3.1.
Applying Lemmas 3.3 and 3.4 and Theorem 3.2, we can obtain the conclusion of Theorem 3.1.
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