In this paper we study the following class of fractional relativistic Schr\"odinger equations: \begin{ equation*} \left\{ \begin{ array}{ ll} (-\Delta+m^{ 2})^{ s}u + V(\varepsilon x) u= f(u) &\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, $s\in (0, 1)$, $m>0$, $N> 2s$, $(-\Delta+m^{ 2})^{ s}$ is the fractional relativistic Schr\"odinger operator, $V:\mathbb{ R}^{ N}\rightarrow \mathbb{ R}$ is a continuous potential satisfying a local condition, and $f:\mathbb{ R}\rightarrow \mathbb{ R}$ is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for $\varepsilon>0$ small enough, the above problem admits a weak solution $u_{ \varepsilon}$ which concentrates around a local minimum point of $V$ as $\varepsilon\rightarrow 0$. We also show that $u_{ \varepsilon}$ has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential $V$ attains its minimum value.
Categories: math.AP