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

Abstract

By using the penalization method and the Ljusternik-Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schrödinger equation $$ \e^{2s}(-Δ)^{s} u + V(x)u = f(u) \mbox{ in } \mathbb{R}^{N} $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ 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 $f(u)=|u|^{q-2}u+λ|u|^{r-2}u$ with $2 0$, by combining a truncation argument and a Moser-type iteration.