Standing waves to upper critical Choquard equation with a local perturbation: multiplicity, qualitative properties and stability

TL;DR

This paper investigates the existence, stability, and qualitative properties of solutions to a critical Choquard equation with local perturbation, extending previous results to a Schrödinger equation context.

Contribution

It provides new results on the existence, stability, and qualitative features of solutions for the upper critical Choquard equation with perturbation, generalizing prior work.

Findings

Existence and orbital stability of ground states.

Positivity, radial symmetry, and exponential decay of second class solutions.

Orbital instability of certain solutions.

Abstract

In this paper, we consider the upper critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -\Delta u=\lambda u+(I_\alpha\ast|u|^{p})|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N),\ \int_{\mathbb{R}^N}|u|^2=a, \end{cases} \end{equation*} where $N\geq 3$, $\mu>0$, $a>0$, $\lambda\in \mathbb{R}$, $\alpha\in (0,N)$, $p=\bar{p}:=\frac{N+\alpha}{N-2}$, $q\in (2,2+\frac{4}{N})$ and $I_\alpha=\frac{C}{|x|^{N-\alpha}}$ with $C>0$. When $\mu a^{\frac{q(1-\gamma_q)}{2}}\leq (2K)^{\frac{q\gamma_q-2\bar{p}}{2(\bar{p}-1)}}$ with $\gamma_q=\frac{N}{2}-\frac{N}{q}$ and $K$ being some positive constant, we prove (1) Existence and orbital stability of the ground states. (2) Existence, positivity, radial symmetry, exponential decay and orbital instability of the ``second class' solutions. This paper generalized and improved parts of the results obtained in \cite{{JEANJEAN-JENDREJ},{Jeanjean-Le},{Soave JFA},{Wei-Wu 2021}} to the Schr\"{o}dinger equation.