Normalized solutions to lower critical Choquard equation in mass-supercritical setting

TL;DR

The paper addresses normalized (mass-constrained) solutions to the nonlocal lower-critical Choquard equation with a mass constraint, introducing a variational framework on the $L^2$-sphere and a novel compactness lemma to overcome lack of compactness from the nonlocal term. It identifies precise regions in the $(c,\\mu)$-plane, via the mixed parameters $\\mathcal{M}_q(c,\\mu)$ and $\\mathcal{N}_{p,q}(c,\\mu)$, where positive radial ground states exist, and derives detailed scaling laws for the Lagrange multiplier and energy. It further establishes sharp asymptotic profiles: as $\\mathcal M_q(c,\\mu) \\to 0$ the normalized ground states converge to the local ground state $U$, while as $\\mathcal M_q(c,\\mu) \\to +\\infty$ they converge to the nonlocal-Hartree extremal $V_{\\delta_0}$; analogous results hold for the power-type nonlinearity and for the related threshold problem $(\\mathcal B_{\\eta})$, including nonexistence and multiplicity phenomena. The results provide new insights into low-regularity, nonlocal, mass-constrained elliptic problems and establish a framework for threshold-based existence and asymptotics in mass-supercritical regimes.

Abstract

We study the normalized solutions to the following Choquard equation \begin{equation*} \aligned &-Δu + λu =μg(u) + γ(I_α* |u|^{\frac{N+α}{N}})|u|^{\frac{N+α}{N}-2}u & \text{in\ \ } \mathbb{R}^N \endaligned \end{equation*} under the $L^2$-norm constraint $\|u\|_2=c$. Here $γ>0$, $ N\geq 1$, $α\in(0,N)$, $I_α$ is the Riesz potential, and the unknown $λ$ appears as a Lagrange multiplier. In a mass supercritical setting on $g$, we find regions in the $(c,μ)$--parameter space such that the corresponding equation admits a positive radial ground state solution. To overcome the lack of compactness resulting from the nonlocal term, we present a novel compactness lemma and some prior energy estimate. These results are even new for the power type nonlinearity $g(u)= |u|^{q-2}u$ with $2+\frac{4}{N}<q<2^*$ ($2^*:=\frac{2N}{N-2}$, if $N\geq 3$ and $2^* = \infty$, if $N=1, 2$). We also show that as $μ$ or $c$ tends to $0$ (resp. $μ$ or $c$ tends to $+\infty$), after a suitable rescaling the ground state solutions converge in $H^1(\RN)$ to a particular solution of the limit equations. Further, we study the non-existence and multiplicity of positive radial solutions to \begin{equation*} -Δu + u = η|u|^{q-2}u + (I_α* |u|^{\frac{N+α}{N}})|u|^{\frac{N+α}{N}-2}u, \quad \text{in}\ \ \RN \end{equation*} where $N \geq 1$, $ 2< q<2^*$ and $η>0$. Based on some analytical ideas the limit behaviors of the normalized solutions, we verify some threshold regions of $η$ such that the corresponding equation has no positive least action solution or admits multiple positive solutions. To the best of our knowledge, this seems to be the first result concerning the non-existence and multiplicity of positive solutions to Choquard type equations involving the lower critical exponent.