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

In this paper, we prove the existence of normalized solutions for thefollowing Schr\"odinger equation \begin{equation*}  \left\{  \begin{array}{ll}  -\Delta u-\lambda u=f(u), & x\in \R^N,  \int_{\R^N}u^2\mathrm{d}x=c  \end{array}  \right.  \end{equation*} with $N\ge3$, $c>0$, $\lambda\in \R$ and $f\in\mathcal{C}(\R,\R)$ in the Sobolev subcritical case with weaker$L^2$-supercritical conditions and in the Sobolev critical case when $f(u)=\mu|u|^{q-2}u+|u|^{2^*-2}u$ with $\mu>0$ and $2<q<2^*=\f{2N}{N-2}$ allowing to be$L^2$-subcritical, critical or supercritical. Our approach is based on severalnew critical point theorems on a manifold, which not only help to weaken theprevious $L^2$-supercritical conditions in the Sobolev subcritical case, butpresent an alternative scheme to construct bounded (PS) sequences on a manifoldwhen $f(u)=\mu |u|^{q-2}u+|u|^{2^*-2}u$ technically simpler than the Ghoussoubminimax principle involving topological arguments, as well as working for all$2<q<2^*$. In particular, we propose new strategies to control the energy levelin the Sobolev critical case which allow to treat, in a unified way, thedimensions $N=3$ and $N\ge 4$, and fulfill what were expected by Soave and byJeanjean-Le . We believe that our approaches and strategies may be adapted andmodified to attack more variational problems in the constraint contexts.

New approaches for Schr\"odinger equations with prescribed mass: The Sobolev subcritical case and The Sobolev critical case with mixed dispersion