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} -\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.