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

In this work we study the following class of problems in $\mathbb R^{N}, N>2s$ $${\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}u+V(z)u=f(u),u(z)>0$$ where $0<s<1$, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian, $\epsilon$ 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 $M\mathrm{.}$ By using variational methods we prove existence, multiplicity and concentration of maxima of positive solutions when $\epsilon \to 0^{+}$. In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set $M$.