Normalized Solutions to Schr\"{o}dinger Equations with Critical Exponent and Mixed Nonlocal Nonlinearities

TL;DR

This paper investigates the existence and nonexistence of normalized solutions to a nonlocal Schrödinger equation with critical and mixed nonlinearities, extending classical problems and analyzing solution behaviors under various parameter regimes.

Contribution

It provides new existence and nonexistence results for normalized solutions in the context of nonlocal Schrödinger equations with critical exponents and mixed nonlinearities, including asymptotic analysis.

Findings

Existence of two solutions for certain parameters with positive and negative energies.

Existence of a normalized ground state for specific parameter ranges and thresholds.

Asymptotic behavior of ground states as the parameter approaches zero.

Abstract

We study the existence and nonexistence of normalized solutions $(u_a, \lambda_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schr\"{o}dinger 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 Schr\"{o}diger equation with a fixed $L^2$-norm $\|u\|_2=a>0$. The leading term is $L^2$-supercritical, that is, $p\in (\frac{N+\alpha+2}{N},\frac{N+\alpha}{N-2}]$, where the Hardy-Littlewood-Sobolev critical exponent $p=\frac{N+\alpha}{N-2}$ appears. We first prove that there exist two normalized solutions if $q\in (\frac{N+\alpha}{N},\frac{N+\alpha+2}{N})$ with $\mu >0$ small, that is, one is at the negative energy level while the other one is at the positive energy level. For $q=\frac{N+\alpha+2}{N}$, we show that there is a normalized ground state for $0<\mu < \tilde{\mu} $ and there exist no ground states for $\mu >\tilde{\mu}$, where $\tilde{\mu}$ is a sharp positive constant. If $q\in (\frac{N+\alpha+2}{N},\frac{N+\alpha}{N-2})$, we deduce that there exists a normalized ground state for any $\mu>0$. We also obtain some existence and nonexistence results for the case $\mu<0$ and $q\in (\frac{N+\alpha}{N},\frac{N+\alpha+2}{N}]$. Besides, we analyze the asymptotic behavior of normalized ground states as $\mu\rightarrow 0^{+}$.