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

In this paper, we study the fractional Schrödinger-Poisson system ( − Δ ) s u + V ( x ) u + K ( x ) ϕ | u | q − 2 u = h ( x ) f ( u ) + | u | 2 s ∗ − 2 u , in R 3 , ( − Δ ) t ϕ = K ( x ) | u | q , in R 3 , $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} (-{\it\Delta})^{s}u+V(x)u+ K(x) \phi|u|^{q-2}u=h(x)f(u)+|u|^{2^{\ast}_{s}-2}u,&\mbox{in}~ {\mathbb R^{3}},\\ (-{\it\Delta})^{t}\phi=K(x)|u|^{q},&\mbox{in}~ {\mathbb R^{3}}, \end{array}\right. \end{array}$$ where s , t ∈ (0, 1), 3 < 4 s < 3 + 2 t , q ∈ (1, 2 s ∗ $\begin{array}{} \displaystyle 2^*_s \end{array}$ /2) are real numbers, (− Δ ) s stands for the fractional Laplacian operator, 2 s ∗ := 6 3 − 2 s $\begin{array}{} \displaystyle 2^{*}_{s}:=\frac{6}{3-2s} \end{array}$ is the fractional critical Sobolev exponent, K , V and h are non-negative potentials and V , h may be vanish at infinity. f is a C 1 -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.

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