Nonlocal Pertubations of Fractional Choquard Equation

TL;DR

This paper investigates the existence of groundstate and sign-changing solutions for a fractional Choquard equation with nonlocal perturbations, using variational methods on the Nehari manifold and nodal sets.

Contribution

It introduces a novel analysis of fractional Choquard equations with nonlocal perturbations, establishing existence results for groundstate and sign-changing solutions.

Findings

Existence of groundstate solutions via minimization on Nehari manifold.

Existence of least energy sign-changing solutions using Nehari nodal set.

Application of variational methods to fractional nonlocal equations.

Abstract

We study the equation \begin{equation} (-\Delta)^{s}u+V(x)u= (I_{\alpha}*|u|^{p})|u|^{p-2}u+\lambda(I_{\beta}*|u|^{q})|u|^{q-2}u \quad\mbox{ in } \R^{N}, \end{equation} where $I_\gamma(x)=|x|^{-\gamma}$ for any $\gamma\in (0,N)$, $p, q >0$, $\alpha,\beta\in (0,N)$, $N\geq 3$ and $ \lambda \in R$. First, the existence of a groundstate solutions using 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.