Ground states for critical fractional Schrödinger‐Poisson systems with vanishing potentials

Abstract

This paper deals with a class of fractional Schrödinger‐Poisson system {array(−Δ)su+V(x)u−K(x)ϕ|u|2s∗−3u=a(x)f(u),arrayx∈ℝ3array(−Δ)sϕ=K(x)|u|2s∗−1,arrayx∈ℝ3$$ \left\{\begin{array}{cc}{\left(-\Delta \right)}^su+V(x)u-K(x)\phi {\left|u\right|}^{2_s^{\ast }-3}u=a(x)f(u),\kern0.30em & x\in {\mathbb{R}}^3\\ {}{\left(-\Delta \right)}^s\phi =K(x){\left|u\right|}^{2_s^{\ast }-1},\kern0.30em & x\in {\mathbb{R}}^3\end{array}\right. $$ with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where s∈false(34,1false),0.1em2s∗=63−2s$$ s\in \left(\frac{3}{4},1\right),{2}_s^{\ast }=\frac{6}{3-2s} $$ is the fractional critical exponent. The problem is set on the whole space, and compactness issues have to be tackled. By employing the mountain pass theorem, concentration‐compactness principle, and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on V$$ V $$, K$$ K $$, a$$ a $$, and f$$ f $$.