[Paper Review] Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schr\"{o}dinger equation

Louis Jeanjean, S.C.-Y. Lu|arXiv (Cornell University)|Nov 25, 2021

Advanced Mathematical Physics ProblemsMathematics34 references23 citations

TL;DR

This paper establishes the existence of normalized solutions with positive energy for a class of coercive nonlinear Schrödinger equations in any dimension N ≥ 1, under general mass-subcritical conditions. By constructing local minimizers and mountain pass critical points on the L2-sphere, it proves that positive energy solutions exist in a left neighborhood of the critical mass m* > 0, even when the global infimum energy is not achieved. The results are applied to the cubic-quintic NLS in R3, revealing that action ground states can be local minimizers or saddle-type solutions, with implications for orbital stability and instability.

ABSTRACT

In any dimension $N \geq 1$, for given mass $m > 0$ and when the $C^1$ energy functional \begin{equation*} I(u) := \frac{1}{2} \int_{\mathbb{R}^N} | abla u|^2 dx - \int_{\mathbb{R}^N} F(u) dx \end{equation*} is coercive on the mass constraint \begin{equation*} S_m := \left\{ u \in H^1(\mathbb{R}^N) ~|~ \|u\|^2_{L^2(\mathbb{R}^N)} = m ight\}, \end{equation*} we are interested in searching for constrained critical points at positive energy levels. Under general conditions on $F \in C^1(\mathbb{R}, \mathbb{R})$ and for suitable ranges of the mass, we manage to construct such critical points which appear as a local minimizer or correspond to a mountain pass or a symmetric mountain pass level. In particular, our results shed some light on the cubic-quintic nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$.

Motivation & Objective

• To identify and construct constrained critical points of the energy functional at positive energy levels for the L2-constrained nonlinear Schrödinger equation.
• To address the lack of systematic study on positive energy solutions in the mass-subcritical regime, where most prior work focuses on negative energy ground states.
• To clarify the variational structure of the cubic-quintic nonlinear Schrödinger equation in R3, particularly the existence and stability of action ground states beyond global minimizers.
• To demonstrate that positive energy solutions can arise as local minimizers or mountain pass levels, even when the global infimum energy is not achieved.

Proposed method

• Introduces a refined geometric estimate (Lemma 2.2 iii) to identify potential positive energy levels on the L2-sphere, based on the behavior of the nonlinearity F near zero.
• Defines a local infimum Em on a subset Sρm of the L2-sphere where the H1-seminorm exceeds a threshold ρ(m*) > 0, ensuring coercivity and compactness.
• Applies variational methods—specifically local minimization and mountain pass theory—on the constrained manifold Sm to find critical points at positive energy levels.
• Uses the mountain pass theorem and symmetric mountain pass theorem to construct solutions when the nonlinearity satisfies a generalized Ambrosetti-Rabinowitz condition (f5).
• Employs concentration-compactness arguments and strong convergence of minimizing sequences (up to translations) to prove existence of minimizers in the local infimum setting.
• Applies the results to the cubic-quintic nonlinearity f(u) = |u|2u - |u|4u in R3, linking the frequency ω to the Lagrange multiplier and analyzing stability via the action functional.

Experimental results

Research questions

• RQ1 Can normalized solutions with positive energy exist for the L2-constrained nonlinear Schrödinger equation in the mass-subcritical regime, even when the global energy infimum is not achieved?
• RQ2 What variational structure (e.g., local minimizer, mountain pass) underlies positive energy solutions in this setting?
• RQ3 How do the properties of the nonlinearity F(u) near zero influence the existence and nature of positive energy solutions?
• RQ4 For the cubic-quintic NLS in R3, can action ground states be local minimizers or saddle-type solutions, and what are their stability properties?

Key findings

• For any m ∈(m**, m*], the local infimum Em > 0 is achieved by a solution v ∈Sm with positive energy and positive Lagrange multiplier, and v has constant sign on RN.
• The local minimizer v is orbitally stable for m ∈(m**, m*], as established via strong convergence of minimizing sequences and standard stability arguments.
• For m > m*, there exist two distinct frequencies ω1, ω2 ∈(0, 3/16) such that the associated action ground states vm and wm satisfy I(vm) = Em > 0 and I(wm) > 0, with wm corresponding to a mountain pass level.
• The action ground state vm = Uω1 is a local but not global minimizer of I|Sm, demonstrating that ground states need not be global minimizers.
• The action ground state wm = Uω2 corresponds to a mountain pass level in the radial subspace H1r(R3), confirming saddle-type behavior.
• The results disprove the converse of [16, Theorem 1.6(i)], showing that a solution with P(u) = 0 does not guarantee a global minimizer, and that m** > 0 is a sharp threshold for existence of positive energy local minimizers.