A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in ℝN

Abstract

In this work we study the following class of problems in (Formula presented.) (Formula presented.) where (Formula presented.), (Formula presented.) is the fractional Laplacian, (Formula presented.) is a positive parameter, the potential (Formula presented.) and the nonlinearity (Formula presented.) satisfy suitable assumptions; in particular it is assumed that (Formula presented.) achieves its positive minimum on some set (Formula presented.) By using variational methods we prove existence and multiplicity of positive solutions when (Formula presented.). In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set (Formula presented.).