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arXiv · 2610.08774

Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass

Daniele Cassani·Giulio Romani
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Abstract

We study Hamiltonian elliptic systems with prescribed bilinear mass $\int_{\mathbb{R}^N}uv=a>0$. 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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Daniele Cassani, Giulio Romani. 2026-10-06. Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass. https://arxiv.org/abs/2610.08774

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