English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the existence and nonexistence of normalized solutions $(u_a,\lambda_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinearSchr\"{o}dinger equation with mixed nonlocal nonlinearities.  This study can be viewed as a counterpart of the Brezis-Nirenberg problem inthe context of normalized solutions to the nonlocal Schr\"{o}diger equationwith 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 theHardy-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 isat the negative energy level while the other one is at the positive energylevel.  For $q=\frac{N+\alpha+2}{N}$, we show that there is a normalized ground statefor $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 thereexists 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$\mu\rightarrow 0^{+}$.

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