Concentration phenomena for a fractional relativistic Schr\"odinger equation with critical growth

Abstract

In this paper, we are concerned with the following fractional relativistic Schr\"odinger equation with critical growth: \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u)+u^{2^{*}_{s}-1} \mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 \, \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent, $(-\Delta+m^{2})^{s}$ is the fractional relativistic Schr\"odinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential $V$, we construct a family of positive solutions $u_{\varepsilon}\in H^{s}(\mathbb{R}^{N})$, with exponential decay, which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$.