Fractional NLS equations with magnetic field, critical frequency and critical growth

Abstract

The paper is devoted to the study of a singularly perturbed fractional Schr\"{o}dinger equations involving critical frequency and critical growth in the presence of a magnetic field. By using variational methods, we obtain the existence of mountain pass solutions $u_{\varepsilon}$ which tend to the trivial solution as $\varepsilon\rightarrow0$. Moreover, we get infinitely many solutions and sign-changing solutions for the problem in absence of magnetic effects under some extra assumptions.