Semiclassical states for fractional Choquard equations with critical growth

Abstract

In this paper, we are concerned with fractional Choquard equation \begin{document}$ \epsilon^{2α}(-Δ)^α u+V(x)u = \epsilon^{μ-3}\Bigl(\int_{\mathbb{R}^3}\frac{|u(y)|^{2_{μ,α}^*}+F(u(y))}{|x-y|^μ}dy\Bigr)\Bigl(|u|^{2_{μ,α}^*-2}u+\frac{1}{2_{μ,α}^*}f(u)\Bigr) {\rm in} \mathbb{R}^3,$ \end{document} where \begin{document} $\epsilon>0$ \end{document} is a parameter, \begin{document} $0 , \begin{document} $0 , \begin{document} $2_{μ,α}^* = \frac{6-μ}{3-2α}$ \end{document} is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator, \begin{document} $f$ \end{document} is a continuous subcritical term, and \begin{document} $F$ \end{document} is the primitive function of \begin{document} $f$ \end{document} . By virtue of the method of Nehari manifold and Ljusternik-Schnirelmann category theory, we prove that the equation has a ground state for \begin{document} $\epsilon$ \end{document} small enough and investigate the relation between the number of solutions and the topology of the set where \begin{document} $V$ \end{document} attains its global minimum for small \begin{document} $\epsilon$ \end{document} . We also obtain sufficient conditions for the nonexistence of ground states.