Multiplicity of normalized solutions to the upper critical fractional Choquard equation with $L^2$-supercritical perturbation

Abstract

We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-\Delta)^s u+V(\varepsilon x)u=\lambda u+\big(I_\alpha*|u|^{p}\big)|u|^{p-2}u+\big(I_\alpha*|u|^{q}\big)|u|^{q-2}u\quad\text{in }\mathbb{R}^N,\] where $N>2s$, $0<s<1$, $(N-4s)^+<\alpha<N$, and the nonlocal exponents satisfy \[\frac{N+2s+\alpha}{N}< q< p=\frac{N+\alpha}{N-2s},\] 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 $\mathrm{cat}_{M_\delta}(M)$ distinct normalized solutions, where $M$ is the set of global minima of $V$ and these solutions concentrate near $M$ as $\varepsilon\to0$.