Multiplicity of normalized solutions to the upper critical fractional Choquard equation with $L^2$-supercritical perturbation
Yergen Aikyn, Yongpeng Chen, Michael Ruzhansky, Zhipeng Yang
TL;DR
This work analyzes normalized solutions with prescribed mass for the upper critical fractional Choquard equation in ℝ^N, featuring dual L^2-supercritical Hartree nonlinearities and a slowly varying potential. The authors develop a constrained variational approach on the L^2-sphere, using a truncation-penalization of the critical term to overcome lack of compactness, and establish a multiplicity result via Lusternik–Schnirelmann category. Under a small-mass condition and for small ε, they prove the existence of at least cat_{M_δ}(M) distinct normalized solutions whose mass concentrates near the global minima set M of V as ε→0; they also analyze the autonomous limit to extract ground-state profiles and demonstrate convergence/concentration behavior. The results extend multiplicity and concentration phenomena to fully L^2-supercritical regimes at the Hartree upper critical exponent for fractional Choquard equations, combining sharp functional-analytic tools with delicate concentration-compactness and category arguments.
Abstract
We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-Δ)^s u+V(\varepsilon x)u=λu+\big(I_α*|u|^{p}\big)|u|^{p-2}u+\big(I_α*|u|^{q}\big)|u|^{q-2}u\quad\text{in }\mathbb{R}^N,\] where $N>2s$, $0<s<1$, $(N-4s)^+<α<N$, and the nonlocal exponents satisfy \[\frac{N+2s+α}{N}< q< p=\frac{N+α}{N-2s},\] so that both nonlinearities are $L^2$-supercritical and the $p$ term has upper critical growth of Hartree type. Under standard assumptions on the slowly varying potential $V$, we develop a constrained variational approach on the $L^2$-sphere, based on a truncation-penalization of the critical term in the energy functional, to overcome the lack of compactness. We prove that, for all sufficiently small $\varepsilon>0$, the problem admits at least $\mathrm{cat}_{M_δ}(M)$ distinct normalized solutions, where $M$ is the set of global minima of $V$ and these solutions concentrate near $M$ as $\varepsilon\to0$.