Weighted Choquard equation perturbed with weighted nonlocal term

Abstract

We investigate the following problem $$ -{\rm div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \Big(|x|^{-θ}*\frac{|u|^{b}}{|x|^α}\Big)\frac{|u|^{b-2}}{|x|^α}u+λ\Big(|x|^{-γ}*\frac{|u|^{c}}{|x|^β}\Big)\frac{|u|^{c-2}}{|x|^β}u \quad\mbox{ in }\R^{N}, $$ where $b,c,\hat{I}\pm ,{\hat{I}}^2>0$, $\hat{I}¸,{\hat{I}}^3\in (0,N)$, $N\ge 3$, $2\le m<\infty$ and $\hat{I}»\in \mathbb{R}$. Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.

19 pages