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 $\varepsilon\rightarrow 0$.
Categories: math.AP