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

TL;DR

This paper establishes new existence results for normalized solutions of Schrödinger equations with prescribed mass in both Sobolev subcritical and critical cases, introducing innovative critical point methods on manifolds.

Contribution

It develops novel critical point theorems on manifolds that weaken supercritical conditions and simplify solution construction for Schrödinger equations with prescribed mass.

Findings

Proved existence of solutions in Sobolev subcritical case with weaker conditions.

Established solutions in Sobolev critical case for mixed dispersion nonlinearities.

Introduced new strategies for energy level control applicable across dimensions.

Abstract

In this paper, we prove the existence of normalized solutions for the following 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 several new critical point theorems on a manifold, which not only help to weaken the previous $L^2$-supercritical conditions in the Sobolev subcritical case, but present an alternative scheme to construct bounded (PS) sequences on a manifold when $f(u)=\mu |u|^{q-2}u+|u|^{2^*-2}u$ technically simpler than the Ghoussoub minimax principle involving topological arguments, as well as working for all $2<q<2^*$. In particular, we propose new strategies to control the energy level in the Sobolev critical case which allow to treat, in a unified way, the dimensions $N=3$ and $N\ge 4$, and fulfill what were expected by Soave and by Jeanjean-Le . We believe that our approaches and strategies may be adapted and modified to attack more variational problems in the constraint contexts.