On a class of nonlinear fractional Schrödinger-Poisson systems

In this paper, we are concerned with the following fractional Schrödinger-Poisson system:\left\{                                   \begin{array}{ll}                                    (-\Delta^s)u+V(x)u+\phi u=m(x)|u|^{q-۲}|u|+f(x,u), & x\in\Omega, \\                                    (-\Delta^t)\phi=u^۲, &  x\in\Omega,\\                                     u=\phi=۰, & x\in\partial\Omega                                   \end{array}                                 \right.where s,t \in (۰,۱], ۲t + ۴s > ۳, ۱ < q < ۲ and \Omega is a bounded smooth domain of \mathbb{R}^۳, and f(x,u) is linearly bounded in u at infinity. Under some assumptions on m, V and f we obtain the existence of non-trivial solutions with the help of the variational methods.

Keywords:

Fractional Schrödinger-Poisson systems , Non-trivial solutions , Variational methods