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 $n=1$), $$\{\begin{array}{llc}(-\Delta {)}^{s}u+{\lambda }_{1}u& =u_1^3+\beta uv,u\in W^{s,2}({\mathbb{R}}^n),(-\Delta {)}^{s}v+{\lambda }_{2}v& =\frac{1}{2}{v}^2+\frac{1}{2}\beta u^2,v\in W^{s,2}({\mathbb{R}}^n),\end{array}$$ where ${\lambda }_j>0$, $j=1,2$, $\beta \in \mathbb{R}$, $n=1,2,3$, and $\frac{n}{4}<s<1$. Precisely, we prove the existence of a positive radially symmetric ground state for any $\beta >0$.

References in corpus (1)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian