Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method

By using the penalization method and the Ljusternik–Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schrödinger equation $$ arepsilon^{2s}(-Delta)^{s} u + V(x)u= f(u) mbox{ in } mathbb{R}^{N} $$ where $arepsilon>0$ is a parameter, $sin (0, 1)$, $N>2s$, $(-Delta)^{s}$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, and $f$ is a superlinear function with subcritical growth. we also obtain a multiplicity result when $f(u)= |u|^{q-2}u + lambda |u|^{r-2}u$ with $2<2^{*}_{s}leq r$ and $lambda >0$, by combining a truncation argument and a Moser-type iteration.