The nonlinear fractional relativistic Schrödinger equation: Existence, multiplicity, decay and concentration results

Abstract

<p style='text-indent:20px;'>In this paper we study the following class of fractional relativistic Schrödinger equations:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u = f(u) &amp;\text{ in } \mathbb{R}^{N}, \\ u\in H^{s}( \mathbb{R}^{N}), \quad u&gt;0 &amp;\text{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon &gt;0 $\end{document}</tex-math></inline-formula> is a small parameter, <inline-formula><tex-math id="M2">\begin{document}$ s\in (0, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ m&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N&gt; 2s $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ (-\Delta+m^{2})^{s} $\end{document}</tex-math></inline-formula> is the fractional relativistic Schrödinger operator, <inline-formula><tex-math id="M6">\begin{document}$ V: \mathbb{R}^{N} \rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a continuous potential satisfying a local condition, and <inline-formula><tex-math id="M7">\begin{document}$ f: \mathbb{R} \rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon &gt;0 $\end{document}</tex-math></inline-formula> small enough, the above problem admits a weak solution <inline-formula><tex-math id="M9">\begin{document}$ u_{\varepsilon } $\end{document}</tex-math></inline-formula> which concentrates around a local minimum point of <inline-formula><tex-math id="M10">\begin{document}$ V $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M11">\begin{document}$ \varepsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. We also show that <inline-formula><tex-math id="M12">\begin{document}$ u_{\varepsilon } $\end{document}</tex-math></inline-formula> has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential <inline-formula><tex-math id="M13">\begin{document}$ V $\end{document}</tex-math></inline-formula> attains its minimum value.</p>