Nonlocal perturbations of the fractional Choquard equation

Abstract

AbstractWe study the equation(-\Delta)^{s}u+V(x)u=(I_{\alpha}*\lvert u\rvert^{p})\lvert u\rvert^{p-2}u+% \lambda(I_{\beta}*\lvert u\rvert^{q})\lvert u\rvert^{q-2}u\quad\text{in }{% \mathbb{R}}^{N},where{I_{\gamma}(x)=\lvert x\rvert^{-\gamma}}for any{\gamma\in(0,N)},{p,q>0},{\alpha,\beta\in(0,N)},{N\geq 3}, and{\lambda\in{\mathbb{R}}}. First, the existence of groundstate solutions by using a minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.