Weighted Choquard equation perturbed with weighted nonlocal term

TL;DR

This paper studies a weighted nonlocal nonlinear PDE involving the Choquard term, focusing on existence of groundstate and sign-changing solutions using variational methods on Nehari manifolds.

Contribution

It introduces new existence results for groundstate and sign-changing solutions to a weighted Choquard equation with nonlocal terms.

Findings

Existence of groundstate solutions established.

Existence of least energy sign-changing solutions demonstrated.

Application of minimization techniques on Nehari manifolds.

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.