Normalized Solutions to Schrödinger Equations with Critical Exponent and Mixed Nonlocal Nonlinearities

Abstract

We study the existence and nonexistence of normalized solutions $(u_a, λ_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schrödinger equation with mixed nonlocal nonlinearities. This study can be viewed as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions to the nonlocal Schrödiger equation with a fixed $L^2$-norm $\|u\|_2=a>0$. The leading term is $L^2$-supercritical, that is, $p\in (\frac{N+α+2}{N},\frac{N+α}{N-2}]$, where the Hardy-Littlewood-Sobolev critical exponent $p=\frac{N+α}{N-2}$ appears. We first prove that there exist two normalized solutions if $q\in (\frac{N+α}{N},\frac{N+α+2}{N})$ with $μ>0$ small, that is, one is at the negative energy level while the other one is at the positive energy level. For $q=\frac{N+α+2}{N}$, we show that there is a normalized ground state for $0<μ< \tildeμ $ and there exist no ground states for $μ>\tildeμ$, where $\tildeμ$ is a sharp positive constant. If $q\in (\frac{N+α+2}{N},\frac{N+α}{N-2})$, we deduce that there exists a normalized ground state for any $μ>0$. We also obtain some existence and nonexistence results for the case $μ<0$ and $q\in (\frac{N+α}{N},\frac{N+α+2}{N}]$. Besides, we analyze the asymptotic behavior of normalized ground states as $μ\rightarrow 0^{+}$.