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

Abstract

AbstractIn this article, we consider the upper critical Choquard equation with a local perturbation−Δu=λu+(Iα∗∣u∣p)∣u∣p−2u+μ∣u∣q−2u,x∈RN,u∈H1(RN),∫RN∣u∣2=a,\left\{\begin{array}{l}-\Delta u=\lambda u+\left({I}_{\alpha }\ast | u\hspace{-0.25em}{| }^{p})| u\hspace{-0.25em}{| }^{p-2}u+\mu | u\hspace{-0.25em}{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1em}{\displaystyle \int }_{{{\mathbb{R}}}^{N}}| u\hspace{-0.25em}{| }^{2}=a,\end{array}\right.whereN≥3N\ge 3,μ>0\mu \gt 0,a>0a\gt 0,λ∈R\lambda \in {\mathbb{R}},α∈(0,N)\alpha \in \left(0,N),p=p¯≔N+αN−2p=\bar{p}:= \frac{N+\alpha }{N-2},q∈2,2+4Nq\in \left(2,2+\frac{4}{N}\right), andIα=C∣x∣N−α{I}_{\alpha }=\frac{C}{| x{| }^{N-\alpha }}withC>0C\gt 0. Whenμaq(1−γq)2≤(2K)qγq−2p¯2(p¯−1)\mu {a}^{\tfrac{q\left(1-{\gamma }_{q})}{2}}\le {\left(2K)}^{\tfrac{q{\gamma }_{q}-2\bar{p}}{2\left(\bar{p}-1)}}withγq=N2−Nq{\gamma }_{q}=\frac{N}{2}-\frac{N}{q}andKKbeing 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 article generalized and improved parts of the results obtained for the Schrödinger equation.