Concentration phenomena for the fractional relativistic Schrödinger–Choquard equation

Abstract

We consider the fractional relativistic Schrödinger–Choquard equation $$\left\{\begin{array}{cc}{(-\mathrm{\Delta }+m^2)}^{s}u+V(𝜀x)u=\left(\frac{1}{|x{|}^{\mu }}\ast F(u)\right)f(u)\hfill & \text{in}{ℝ}^N,\hfill \\ u\in H^s({ℝ}^N),\hfill & u>0\text{in}{ℝ}^N,\hfill \end{array}\right.$$ where $𝜀>0$ is a small parameter, $s\in (0,1)$, $m>0$, $N>2s$, $\mu \in (0,2s)$, ${(-\mathrm{\Delta }+m^2)}^s$ is the fractional relativistic Schrödinger operator, $V:{ℝ}^N\to ℝ$ is a continuous potential having a local minimum, $f:ℝ\to ℝ$ is a continuous nonlinearity with subcritical growth at infinity and $F(t)={\int }_0^{t}f(\tau )d\tau$. Exploiting appropriate variational arguments, we construct a family of solutions concentrating around the local minimum of $V$ as $𝜀\to 0$.