New approaches for Schrödinger equations with prescribed mass: The Sobolev subcritical case and The Sobolev critical case with mixed dispersion

TL;DR

The paper addresses the existence of normalized solutions to nonlinear Schrödinger equations with a prescribed $L^2$-mass, covering both Sobolev subcritical and Sobolev critical regimes with mixed dispersion. It introduces new critical-point theories on constraint manifolds to construct bounded PS sequences and weakens $L^2$-supercritical hypotheses, providing a unified framework beyond Ghoussoub’s topological minimax. A key contribution is the development of energy-estimation techniques and testing function constructions that control the Sobolev-critical energy barrier, enabling compactness and convergence across dimensions $N=3$ and $N\ge4$ and for all $2<q<2^*$. The results yield multiple normalized solutions, including ground states and mountain-pass type solutions, and the methods promise applicability to a broader class of constrained variational problems in nonlinear PDEs.

Abstract

In this paper, we prove the existence of normalized solutions for the following Schrödinger equation \begin{equation*} \left\{ \begin{array}{ll} -Δu-λu=f(u), & x\in \R^N, \int_{\R^N}u^2\mathrm{d}x=c \end{array} \right. \end{equation*} with $N\ge3$, $c>0$, $λ\in \R$ 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 $f(u)=μ|u|^{q-2}u+|u|^{2^*-2}u$ with $μ>0$ 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 $f(u)=μ|u|^{q-2}u+|u|^{2^*-2}u$ 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.

Theorems & Definitions (54)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof