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Abstract
In this paper, we are concerned with the problem driven by a non-local integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study multiple solutions for the following non-local fractional Laplace equations:where
is fixed parameter,
is an open bounded subset of
with smooth boundary
(
) and
is the fractional Laplace operator. By a variant version of the Mountain Pass Theorem, a multiplicity result is obtained for the above-mentioned superlinear problem without Ambrosetti–Rabinowitz condition. Consequently, the result may be looked as a complete extension of the previous work of Wang and Tang to the non-local fractional setting.
Acknowledgements
The authors are indebted to the referees for helpful comments and suggestions. B. Zhang was supported by the Natural Science Foundation of Heilongjiang Province of China (No. A201306), the Doctoral Research Foundation of Heilongjiang Institute of Technology (No. 2013BJ15) and the Research Foundation of Heilongjiang Educational Committee (No. 12541667).