On inequalities of Bliss-Moser type with loss of compactness in $\mathbb{R}^N$

TL;DR

This paper establishes optimal limiting Bliss inequalities in  with loss of compactness at critical parameters, extending previous results to higher dimensions and analyzing the sharpness of the inequalities.

Contribution

It extends Bliss inequalities with loss of compactness from 2D to higher dimensions and proves their optimality and criticality.

Findings

Inequalities are optimal for eta ; no further improvements.

Existence of compactness for eta < 1, loss at eta = 1.

Extension of previous 2D results to general dimensions N .

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^{\beta\left(\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,\beta), \ \hbox{ for } \beta \le 1 \end{equation} The inequalities are optimal with respect to $\beta \le 1$; there is compactness for $\beta<1$, and along the infinitesimal Moser sequence for $\beta = 1$. 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}+\gamma\log\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,\gamma) \end{equation} hold for $\gamma\leq1$, and for $\gamma=1$ 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 \'{O}, 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 $N\geq2$.