Bound state solutions for non-autonomous fractional Schrödinger-Poisson equations with critical exponent

Abstract

In this paper, we study the fractional Schrödinger-Poisson equation \begin{equation*} \ \left\{\begin{aligned} &(-Δ)^{s}u+V(x)u+K(x)ϕu=|u|^{2^{\ast}_{s}-2}u, &\mbox{in} \ \mathbb{R}^{3},\\ &(-Δ)^{s}ϕ=K(x)u^{2},&\mbox{in} \ \mathbb{R}^{3}, \end{aligned}\right. \end{equation*} where $s\in (\frac{3}{4},1]$, $2_s^{\ast }=\frac{6}{3-2s}$ is the fractional critical exponent, $K\in L^{\frac{6}{6s-3}}({\mathbb{R}}^3)$ and $V\in L^{\frac{3}{2s}}({\mathbb{R}}^3)$ are nonnegative functions. If $\Vert V{\Vert }_{\frac{3}{2s}}+\Vert K{\Vert }_{\frac{6}{6s-3}}$ is sufficiently small, we prove that the equation has at least one bound state solution.

References in corpus (2)

• Fractional Quantum Mechanics
• Nonlinear fractional Schrödinger equations in one dimension