Ground states of some coupled nonlocal fractional dispersive PDEs

Abstract

We show the existence of ground state solutions to the following stationary system coming from some coupled fractional dispersive equations such as: nonlinear fractional Schrödinger (NLFS) equations (for dimension $n=1,\, 2,\, 3$) or NLFS and fractional Korteweg-de Vries equations (for $n=1$), $$ \left \{ \begin{array}{ll} (-Δ)^{s} u+ λ_1 u &= u_1^{3}+βuv,\quad u\in W^{s,2}(\mathbb{R}^n), (-Δ)^{s} v + λ_2 v &= \frac 12 v^{2}+\frac 12 βu^2,\quad v\in W^{s,2}(\mathbb{R}^n), \end{array} \right. $$ where $λ_j>0$, $j=1,2$, $β\in \mathbb{R}$, $n=1,\, 2,\, 3$, and $\frac n4< s<1$. Precisely, we prove the existence of a positive radially symmetric ground state for any $β>0$.