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

Abstract

In this paper, we consider the upper critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -Δu=λu+(I_α\ast|u|^{p})|u|^{p-2}u+μ|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 $N\ge 3$, $\mu >0$, $a>0$, $\lambda \in \mathbb{R}$, $\alpha \in (0,N)$, $p=\bar{p}:=\frac{N+\alpha }{N-2}$, $q\in (2,2+\frac{4}{N})$ and $I_{\alpha }=\frac{C}{|x{|}^{N-\alpha }}$ with $C>0$. When $\mu a^{\frac{q(1-{\gamma }_q)}{2}}\le (2K{)}^{\frac{q{\gamma }_q-2\bar{p}}{2(\bar{p}-1)}}$ 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 Schrödinger equation.

arXiv admin note: text overlap with arXiv:2103.07026