Nonexistence, existence and symmetry of normalized ground states to Choquard equations with a local perturbation

TL;DR

This paper investigates the existence, nonexistence, and symmetry of normalized ground states for a class of Choquard equations with local perturbations, covering critical cases and employing variational and rearrangement techniques.

Contribution

It provides a comprehensive analysis of ground states for Choquard equations with local perturbations, including critical cases, using advanced mathematical methods.

Findings

Established conditions for nonexistence of solutions.

Proved existence and symmetry of ground states under certain parameters.

Extended results to Hardy-Littlewood-Sobolev critical exponent case.

Abstract

We study the Choquard equation with a local perturbation \begin{equation*} -\Delta u=\lambda u+(I_\alpha\ast|u|^p)|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2. \end{equation*} For a $L^2$-critical or $L^2$-supercritical perturbation $\mu|u|^{q-2}u$, we prove nonexistence, existence and symmetry of normalized ground states, by using the mountain pass lemma, the Poho\v{z}aev constraint method, the Schwartz symmetrization rearrangements and some theories of polarizations. In particular, our results cover the Hardy-Littlewood-Sobolev upper critical exponent case $p=(N+\alpha)/(N-2)$. Our results are a nonlocal counterpart of the results in \cite{{Li 2021-4},{Soave JFA},{Wei-Wu 2021}}.