规范化哈密顿椭圆系统：双线性质量的Gagliardo-Nirenberg不等式
Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass
- Università degli Studi dell’Insubria（因苏布里亚大学）
- RISM-Riemann International School of Mathematics（RISM-黎曼国际数学学院）
- Università degli Studi di Udine（乌迪内大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究具有规定双线性质量的哈密顿椭圆系统，建立Gagliardo-Nirenberg不等式，分类正规范化解，并发展指数非线性理论，获得最小能量径向解及质量响应公式。
AI中文摘要：
我们研究具有规定双线性质量 $\int_{\mathbb{R}^N}uv=a>0$ 的哈密顿椭圆系统。我们在交叉梯度配对和双线性质量下，在尺度不变的组件有界类上建立了Gagliardo-Nirenberg不等式。它确定了临界曲线 $1/p+1/q=N/(N+2)$ 和受限能量三分法。对于幂非线性，我们在Sobolev次临界双曲线上建立了正解的唯一性（模公共平移）和非退化性。精确缩放随后对正规范化解进行分类，并确定唯一的临界质量。在二维情形下，我们发展了具有尖锐梯度阈值和最优四次首项系数的截断标量精化的双线性指数Gagliardo-Nirenberg估计。借鉴[Cassani-Tarsi, this http URL (2015)]的变分思想，我们结合质量归一化商、在完全负纤维上的约化以及低于浓度阈值的紧性，对于纯指数非线性，在低于三次极限质量的每个规定质量下获得最小能量正径向解。局部微扰分支和小频率指数分支产生显式质量响应公式。
英文摘要：
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.
