Multi-bump solutions for fractional Schrödinger equation with electromagnetic fields and critical nonlinearity

Abstract

In this article, we consider the existence of multi-bump solutions for a class of the fractional Schrödinger equation with external magnetic field and critical nonlinearity in $\mathbb{R}^N$: $$(-\Delta)_A^su + (\lambda V(x) + Z(x))u = \beta f(|u|^2)u + |u|^{2_s^\ast-2}u,$$ where $f$ is a continuous function satisfying Ambrosetti-Rabinowitz condition, and $V: \mathbb{R}^N \rightarrow\mathbb{R}$ has a potential well $\Omega := \mbox{int}V^{-1}(0)$ which possesses $k$ disjoint bounded components $\Omega := \cup_{j=1}^k\Omega_j$. By using variational methods, we prove that if the parameter $\lambda > 0$ is large enough, then the equation has at least $2^k-1$ multi-bump solutions.