Weighted Choquard equation perturbed with weighted nonlocal term

Abstract: We investigate the following problem $$ -{\rm div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \Big(|x|^{-\theta}\frac{|u|^{b}}{|x|^{\alpha}}\Big)\frac{|u|^{b-2}}{|x|^{\alpha}}u+\lambda\Big(|x|^{-\gamma}\frac{|u|^{c}}{|x|^{\beta}}\Big)\frac{|u|^{c-2}}{|x|^{\beta}}u \quad\mbox{ in }\R^{N}, $$ where $b, c, \alpha, \beta >0$, $\theta,\gamma \in (0,N)$, $N\geq 3$, $2\leq m< \infty$ and $\lambda \in \R$. 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.