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
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).
2. Abstract Framework and Fractional Sobolev Spaces
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.
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 .
3. Main Theorem
and we assume that
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:
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 .
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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