Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass
Abstract
We study Hamiltonian elliptic systems with prescribed bilinear mass ∫ℝNuv=a>0. We establish a Gagliardo--Nirenberg inequality in the crossed gradient pairing and bilinear mass, on scaling-invariant component-bounded classes. It identifies the critical curve 1/p+1/q=N/(N+2) and the restricted energy trichotomy. For power nonlinearities, we establish uniqueness up to common translations and nondegeneracy of positive profiles throughout the Sobolev-subcritical hyperbola. Exact scaling then classifies positive normalised solutions and identifies the unique critical mass. In dimension two, we develop a bilinear exponential Gagliardo--Nirenberg estimate with a sharp gradient threshold and a truncated scalar refinement with the optimal quartic leading coefficient. Adapting variational ideas from [Cassani-Tarsi, Calc.Var.PDE (2015)] we combine a mass-normalising quotient, reduction over the full negative fibres and compactness below the concentration threshold to obtain least-energy positive radial solutions for pure exponential nonlinearities at every prescribed mass below the cubic limiting mass. Local perturbative branches and a small-frequency exponential branch yield explicit mass-response formulas.
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BibTeX
@misc{cassani2026normalised,
title = {{Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass}},
author = {Daniele Cassani and Giulio Romani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08774},
eprint = {2610.08774},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08774},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass
AU - Daniele Cassani
AU - Giulio Romani
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.08774
UR - https://arxiv.org/abs/2610.08774
AB - We study Hamiltonian elliptic systems with prescribed bilinear mass ∫_ℝ^Nuv=a>0. We establish a Gagliardo--Nirenberg inequality in the crossed gradient pairing and bilinear mass, on scaling-invariant component-bounded classes. It identifies the critical curve 1/p+1/q=N/(N+2) and the restricted energy trichotomy. For power nonlinearities, we establish uniqueness up to common translations and nondegeneracy of positive profiles throughout the Sobolev-subcritical hyperbola. Exact scaling then classifies positive normalised solutions and identifies the unique critical mass. In dimension two, we develop a bilinear exponential Gagliardo--Nirenberg estimate with a sharp gradient threshold and a truncated scalar refinement with the optimal quartic leading coefficient. Adapting variational ideas from [Cassani-Tarsi, Calc.Var.PDE (2015)] we combine a mass-normalising quotient, reduction over the full negative fibres and compactness below the concentration threshold to obtain least-energy positive radial solutions for pure exponential nonlinearities at every prescribed mass below the cubic limiting mass. Local perturbative branches and a small-frequency exponential branch yield explicit mass-response formulas.
ER -