| Summary: | In this article, we study the following fractional Schrödinger-Poisson system: ε2s(−Δ)su+V(x)u+ϕu=f(u)+∣u∣2s*−2u,inR3,ε2t(−Δ)tϕ=u2,inR3,\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<1,2s+2t>30\lt s,t\lt 1,2s+2t\gt 3, and 2s*=63−2s{2}_{s}^{* }=\frac{6}{3-2s} is the critical Sobolev exponent in dimension 3. By assuming that VV 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 ε0>0{\varepsilon }_{0}\gt 0 such that for all ε∈(0,ε0]\varepsilon \in (0,{\varepsilon }_{0}], the above system has a semiclassical Nehari-Pohozaev-type ground state solution vˆε{\hat{v}}_{\varepsilon }. Moreover, the decay estimate and asymptotic behavior of {vˆε}\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.
|
|---|