Three positive solutions for the indefinite fractional Schrödinger-Poisson systems - Che, Guofeng; Wu, Tsung-fang (Topological Methods in Nonlinear Analysis, 2023)
In this paper, we are concerned with the following fractionalSchrödinger-Poisson systems with concave-convex nonlinearities:\begin{equation*}\begin{cases}(-\Delta )^{s}u+u+\mu l(x)\phi u=f(x)|u|^{p-2}u+g(x)|u|^{q-2}u& \text{in }\mathbb{R}^{3}, \\(-\Delta )^{t}\phi =l(x)u^{2} & \text{in }\mathbb{R}^{3},%\end{cases}\end{equation*}where ${1}/{2}< t\leq s< 1$, $1< q< 2< p< \min \{4,2_{s}^{\ast }\}$,$2_{s}^{\ast }={6}/({3-2s})$, and $\mu > 0$ is a parameter,$f\in C\big(\mathbb{R}^{3}\big)$ is sign-changing in $\mathbb{R}^{3}$and $g\in L^{p/(p-q)}\big(\mathbb{R}^{3}\big)$. Under some suitable assumptions on $l(x)$, $f(x)$ and $g(x)$, weexplore that the energy functional corresponding to the system is coerciveand bounded below on $H^{\alpha }\big(\mathbb{R}^{3}\big)$ which gets a positivesolution. Furthermore, we constructed some new estimation techniques, and obtained othertwo positive solutions. Recent results from the literatureare generally improved and extended.