A multiplicity result via Ljusternick-Schnirelmann category and morse theory for a fractional schrödinger equation in $\mathbb R^{N}$

Abstract

In this work we study the following class of problems in $\mathbb R^{N}, N>2s$ $$ \varepsilon^{2s} (-Δ)^{s}u + V(z)u=f(u), \,\,\, u(z) > 0 $$ where $0<s<1$, $(-Δ)^{s}$ is the fractional Laplacian, $\varepsilon$ is a positive parameter, the potential $V:\mathbb{R}^N \to\mathbb{R}$ %is a continuous functions 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, multiplicity and concentration of maxima 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 $M$.