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

Abstract

In this paper we study the following class of fractional relativistic Schrödinger equations: \begin{equation*} \left\{ \begin{array}{ll} (-Δ+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 Schrödinger operator, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous potential satisfying a local condition, and $f:\mathbb{R}\to \mathbb{R}$ 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.

References in corpus (1)

• Concentration phenomena for a class of fractional Kirchhoff equations in ${\mathbb{R}}^N$ with general nonlinearities

Cited by in corpus (1)

• Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth