Concentration phenomenon for a fractional Schrödinger equation with discontinuous nonlinearity

Abstract

We study the following fractional Schrödinger equation with discontinuous nonlinearity: \begin{document}$ \begin{align*} \left\{ \begin{array}{ll} (-\Delta)^{s}u + V( \varepsilon x) u = H(u-\beta)f(u) \mbox{ in } \mathbb{R}^{N}, \\ u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{align*} $\end{document} where $ \varepsilon, \beta>0 $, $ s\in (0, 1) $, $ N> 2s $, $ H $ is the Heaviside function, $ (-\Delta)^{s} $ is the fractional Laplacian operator, $ V:\mathbb{R}^{N} \rightarrow \mathbb{R} $ is a continuous potential satisfying del Pino-Felmer type assumptions and $ f:\mathbb{R}\rightarrow \mathbb{R} $ is a superlinear continuous nonlinearity with subcritical growth at infinity. By using a penalization method and nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem.