Existence of ground state solutions for an asymptotically 2-linear fractional Schrödinger–Poisson system

Abstract

AbstractIn this paper, we investigate the following fractional Schrödinger–Poisson system: $$\left \{ \textstyle\begin{array}{l@{\quad}l} (-\Delta)^{s} u + u + \phi u = f(u), & \text{in } \mathbb{R}^{3}, \\ (-\Delta)^{t} \phi= u^{2}, & \text{in } \mathbb{R}^{3}, \end{array}\displaystyle \right . $${(−Δ)su+u+ϕu=f(u),in R3,(−Δ)tϕ=u2,in R3, where $\frac{3}{4} < s < 1$34<s<1, $\frac{1}{2} < t < 1$12<t<1, and f is a continuous function, which is superlinear at zero, with $f(\tau) \tau \ge3 F(\tau) \ge0$f(τ)τ≥3F(τ)≥0, $F(\tau) = \int_{0}^{\tau} f(s) \,ds$F(τ)=∫0τf(s)ds, $\tau \in\mathbb{R}$τ∈R. We prove that the system admits a ground state solution under the asymptotically 2-linear condition. The result here extends the existing study.