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 $\epsilon >0$ is a parameter, $s\in (0,1)$, $N>2s$, $(-\Delta {)}^s$ 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 $2<q<2_s^{\ast }\le r$ and $\lambda >0$, by combining a truncation argument and a Moser-type iteration.

Cited by in corpus (1)

• Multiplicity and concentration results for a fractional Choquard equation via penalization method