PreprintCC BY 4.0arXiv
An alternative variational approach to constrained critical points for nonlinear Schrödinger equations with combined nonlinearities
Abstract
We introduce a new variational framework for the study of standing waves for nonlinear Schrödinger equations with combined power nonlinearities. The approach is based on minimizing the mass functional over a set defined by an upper bound on the energy and a lower bound on a nonlinear potential term, rather than minimizing the energy at prescribed mass. This yields a new characterization of the sharp Gagliardo--Nirenberg optimizer through purely variational constraints. As a consequence, we establish the existence of minimizers for all energy levels below a critical threshold and derive the corresponding Euler--Lagrange equations. These minimizers give rise to standing-wave solutions of the nonlinear Schröedinger equation
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BibTeX
@misc{bellazzini2026alternative,
title = {{An alternative variational approach to constrained critical points for nonlinear Schrödinger equations with combined nonlinearities}},
author = {Jacopo Bellazzini and Pietro Majer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37814},
eprint = {2610.37814},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37814},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An alternative variational approach to constrained critical points for nonlinear Schrödinger equations with combined nonlinearities
AU - Jacopo Bellazzini
AU - Pietro Majer
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37814
UR - https://arxiv.org/abs/2610.37814
AB - We introduce a new variational framework for the study of standing waves for nonlinear Schrödinger equations with combined power nonlinearities. The approach is based on minimizing the mass functional over a set defined by an upper bound on the energy and a lower bound on a nonlinear potential term, rather than minimizing the energy at prescribed mass. This yields a new characterization of the sharp Gagliardo--Nirenberg optimizer through purely variational constraints. As a consequence, we establish the existence of minimizers for all energy levels below a critical threshold and derive the corresponding Euler--Lagrange equations. These minimizers give rise to standing-wave solutions of the nonlinear Schröedinger equation
ER -