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Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity (1709.08207v1)
Published 24 Sep 2017 in math.AP
Abstract: In this paper we focus our attention on the following nonlinear fractional Schr\"odinger equation with magnetic field \begin{equation*} \varepsilon{2s}(-\Delta)_{A/\varepsilon}{s}u+V(x)u=f(|u|{2})u \quad \mbox{ in } \mathbb{R}{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N\geq 3$, $(-\Delta){s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}{N}\rightarrow \mathbb{R}$ and $A:\mathbb{R}{N}\rightarrow \mathbb{R}N$ are continuous potentials and $f:\mathbb{R}{N}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick-Schnirelmann theory, we prove existence and multiplicity of solutions for $\varepsilon$ small.