On inequalities of Bliss-Moser type with loss of compactness in $\mathbb{R}^N$
Yanyan Guo, Huxiao Luo, Bernhard Ruf
TL;DR
The paper analyzes sharp Bliss-type inequalities with loss of compactness in $\mathbb{R}^N$ at two limiting scales. It proves a Limiting Bliss inequality with exponent $\beta(\log\frac{e}{s})$ in a one-dimensional reduction, establishing finiteness for $\beta\le 1$ and compactness for $\beta<1$, with loss along the infinitesimal Moser sequence at $\beta=1$. It also proves a log–log improved version with exponent $\log\frac{e}{s}+\gamma\log\log\frac{e}{s}$ finite for $\gamma\le 1$ and critical for $\gamma=1$, with no further improvement of the exponent possible. The results extend a recent $N=2$ DRUUbilla-type finding to general dimensions $N\ge 2$, and the analysis hinges on Bliss embedding constants, Moser-type sequences, and a detailed theory of approximate Moser functions to control concentration and non-compactness.
Abstract
We prove the following Limiting Bliss inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{β\left(\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,β), \ \hbox{ for } β\le 1 \end{equation} The inequalities are optimal with respect to $β\le 1$; there is compactness for $β<1$, and along the infinitesimal Moser sequence for $β= 1$. Moreover, we show that the improved inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{\left(\log\frac{e}{s}+γ\log\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,γ) \end{equation} hold for $γ\leq1$, and for $γ=1$ the inequalities are critical with loss of compactness. The inequalities are optimal: no further improvement in the coefficient of the exponent is possible. The second result extends the result in [J. M. do Ó, B. Ruf and P. Ubilla, A critical Moser type inequality with loss of compactness due to infinitesimal shocks, Calc. Var. Partial Differential Equations 62 (2023)] from $N=2$ to general dimensions $N\geq2$.