Document Type : Special issue editorial

Authors

• M. Soluki ¹
• S.H. Rasouli ²
• G.A. Afrouzi ¹

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

where $s,t\in (0,1],2t+4s>3,1<q<2$ and $\mathrm{\Omega }$ 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

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