On a class of nonlinear fractional Schrödinger-Poisson systems

Document Type : Special issue editorial

Authors

1 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

2 Department of Mathematics, Faculty of Basic Sciences, Babol (Noushirvani) University of Technology Babol, Iran

Abstract

In this paper, we are concerned with the following fractional Schrödinger-Poisson system:

$$\left\{
                                   \begin{array}{ll}
                                    (-\Delta^s)u+V(x)u+\phi u=m(x)|u|^{q-2}|u|+f(x,u), & x\in\Omega, \\
                                    (-\Delta^t)\phi=u^2, &  x\in\Omega,\\
                                     u=\phi=0, & x\in\partial\Omega
                                   \end{array}
                                 \right.$$

where $s,t \in (0,1], 2t + 4s > 3, 1 < q < 2$ and $\Omega$ is a bounded smooth domain of $\mathbb{R}^3$, 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