Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials

Abstract

AbstractIn this paper, we study the fractional Schrödinger-Poisson system(−Δ)su+V(x)u+K(x)ϕ|u|q−2u=h(x)f(u)+|u|2s∗−2u,in R3,(−Δ)tϕ=K(x)|u|q,in R3,$$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} (-{\it\Delta})^{s}u+V(x)u+ K(x) \phi|u|^{q-2}u=h(x)f(u)+|u|^{2^{\ast}_{s}-2}u,&\mbox{in}~ {\mathbb R^{3}},\\ (-{\it\Delta})^{t}\phi=K(x)|u|^{q},&\mbox{in}~ {\mathbb R^{3}}, \end{array}\right. \end{array}$$wheres,t∈ (0, 1), 3 < 4s< 3 + 2t,q∈ (1,2s∗$\begin{array}{} \displaystyle 2^*_s \end{array}$/2) are real numbers, (−Δ)sstands for the fractional Laplacian operator,2s∗:=63−2s$\begin{array}{} \displaystyle 2^{*}_{s}:=\frac{6}{3-2s} \end{array}$is the fractional critical Sobolev exponent,K,Vandhare non-negative potentials andV,hmay be vanish at infinity.fis aC1-function satisfying suitable growth assumptions. We show that the above fractional Schrödinger-Poisson system has a positive and a sign-changing least energy solution via variational methods.