Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions

Abstract

Abstract In this article, we study the following fractional Schrödinger-Poisson system: ε 2 s ( − Δ ) s u + V ( x ) u + ϕ u = f ( u ) + ∣ u ∣ 2 s * − 2 u , in R 3 , ε 2 t ( − Δ ) t ϕ = u 2 , in R 3 , \left\{\begin{array}{ll}{\varepsilon }^{2s}{\left(-\Delta )}^{s}u+V\left(x)u+\phi u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{2t}{\left(-\Delta )}^{t}\phi ={u}^{2},\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right. where ε > 0 \varepsilon \gt 0 is a small parameter, 0 < s , t < 1 , 2 s + 2 t > 3 0\lt s,t\lt 1,2s+2t\gt 3 , and 2 s * = 6 3 − 2 s {2}_{s}^{* }=\frac{6}{3-2s} is the critical Sobolev exponent in dimension 3. By assuming that V V is weakly differentiable and f ∈ C ( R , R ) f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) satisfies some lower order perturbations, we show that there exists a constant ε 0 > 0 {\varepsilon }_{0}\gt 0 such that for all ε ∈ ( 0 , ε 0 ] \varepsilon \in (0,{\varepsilon }_{0}] , the above system has a semiclassical Nehari-Pohozaev-type ground state solution v ˆ ε {\hat{v}}_{\varepsilon } . Moreover, the decay estimate and asymptotic behavior of { v ˆ ε } \left\{{\hat{v}}_{\varepsilon }\right\} are also investigated as ε → 0 \varepsilon \to 0 . Our results generalize and improve the ones in Liu and Zhang and Ambrosio, and some other relevant literatures.