Infinitely many Solutions with Peaks for a Fractional System in ℝN

Abstract

In this article, we consider the following coupled fractional nonlinear Schrodinger system in ℝN$$\begin{cases}(-\Delta)^s u+P(x)u=\mu _1 |u|^{2p-2}u+\beta|v|^p|u|^{p-2}u, & x \in \mathbb{R}^N, \\ (-\Delta)^s v+Q(x)v=\mu_2|v|^{2p-2}v+\beta|u|^p|v|^{p-2}v, & x \in \mathbb{R}^N, \\ u, v \in H^s(\mathbb{R}^N), \end{cases}$$ where $$N \ge 2,0 0,\mu _{2} > 0$$ and β ϵ ℝ is a coupling constant. We prove that it has infinitely many non-radial positive solutions under some additional conditions on P(x), Q(x), p and β. More precisely, we will show that for the attractive case, it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that P(x) and Q(x) satisfy some algebraic decay at infinity.