The nonlinear fractional relativistic Schr\"odinger equation: existence, multiplicity, decay and concentration results

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 $\epsilon >0$ is a small parameter, $s\in (0, 1)$,$m>0$,$N> 2s$,$(-\Delta+m^{2})^{s}$ is the fractional relativistic Schr"odinger operator, $V:{\mathbb{R}}^N\to \mathbb{R}$ 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 $\epsilon >0$ small enough, the above problem admits a weak solution $u_{\epsilon }$ which concentrates around a local minimum point of $V$ as $\epsilon \to 0$. We also show that $u_{\epsilon }$ 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.