The nonlinear fractional relativistic Schrödinger 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 \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $(-\Delta+m^{2})^{s}$ operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ continuous potential satisfying local condition, and $f:\mathbb{R}\rightarrow subcritical nonlinearity. By using variant extension method penalization technique, first prove that, for enough, above problem admits weak solution $u_{\varepsilon}$ which concentrates around minimum point $V$ as $\varepsilon\rightarrow 0$. We also show that has an exponential decay at infinity by constructing suitable comparison function performing some refined estimates. Secondly, combining generalized Nehari manifold Ljusternik-Schnirelman theory, relate number positive solutions with topology set attains its value.