Ground states of some coupled nonlocal fractional dispersive PDEs (1801.09595v2)

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 Schr\"odinger (NLFS) equations (for dimension $n=1,\, 2,\, 3$) or NLFS and fractional Korteweg-de Vries equations (for $n=1$), $$ \left { \begin{array}{ll} (-\Delta)^{s} u+ \lambda_1 u &= u_1^{3}+\beta uv,\quad u\in W^{s,2}(\mathbb{R}^n), (-\Delta)^{s} v + \lambda_2 v &= \frac 12 v^{2}+\frac 12 \beta u^2,\quad v\in W^{s,2}(\mathbb{R}^n), \end{array} \right. $$ where $\lambda_j>0$, $j=1,2$, $\beta\in \mathbb{R}$, $n=1,\, 2,\, 3$, and $\frac n4< s<1$. Precisely, we prove the existence of a positive radially symmetric ground state for any $\beta>0$.