Abstract
In this work we study the following class of problems in \({\mathbb R^{N}, N > 2s}\)
where \({0 < s < 1}\), \({(-\Delta)^{s}}\) is the fractional Laplacian, \({\varepsilon}\) is a positive parameter, the potential \({V : \mathbb{R}^N \to \mathbb{R}}\) and the nonlinearity \({f : \mathbb R \to \mathbb R}\) satisfy suitable assumptions; in particular it is assumed that \({V}\) achieves its positive minimum on some set \({M.}\) By using variational methods we prove existence and multiplicity of positive solutions when \({\varepsilon \to 0^{+}}\). In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set \({M}\).
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Giovany M. Figueiredo was partially supported by CNPq/Brazil. Gaetano Siciliano was partially supported by Fapesp and CNPq, Brazil.
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Figueiredo, G.M., Siciliano, G. A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in \({\mathbb R^{N}}\) . Nonlinear Differ. Equ. Appl. 23, 12 (2016). https://doi.org/10.1007/s00030-016-0355-4
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DOI: https://doi.org/10.1007/s00030-016-0355-4