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

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^{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 $\beta \le 1$; 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}+γ\log\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,γ) \end{equation} hold for $\gamma \le 1$, 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 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 $N\ge 2$.