A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in \({\mathbb R^{N}}\)

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Abstract

In this work we study the following class of problems in \({\mathbb R^{N}, N > 2s}\)
$$\varepsilon^{2s}(-\Delta)^{s}u + V(z)u = f(u), \,\,\,u(z) > 0$$
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 \({M}\).

Keywords

Fractional Laplacian multiplicity of solutions Ljusternick-Schnirelmann category Morse theory 

Mathematics Subject Classification

35A15 35S05 58E05 74G35 
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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Faculdade de MatemáticaUniversidade Federal do ParáBelémBrazil
  2. 2.Departamento de Matemática, Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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