Maximizers for the singular Trudinger-Moser inequalities in the subcritical cases

Abstract:

The main purpose of this note is to study the existence of extremal functions for the singular Trudinger-Moser inequalities in the subcritical cases. More precisely, let $N\geq 2$,$~0<\beta <N,~0<a,~b$ and denote \begin{align*} TM_{a,b,\beta }\left ( \alpha \right ) & =\sup _{\left \Vert \nabla u\right \Vert _{N}^{a}+\left \Vert u\right \Vert _{N}^{b}\leq 1}\int _{ \mathbb {R}^{N}}\phi _{N}\left ( \alpha \left ( 1-\frac {\beta }{N}\right ) \left \vert u\right \vert ^{\frac {N}{N-1}}\right ) \frac {dx}{\left \vert x\right \vert ^{\beta }},\\ \phi _{N}(t) & =e^{t}- {\displaystyle \sum \limits _{j=0}^{N-2}} \frac {t^{j}}{j!}. \end{align*} Then we will prove in this article that $TM_{a,b,\beta }\left ( \alpha \right )$ can be attained if ($\alpha <\alpha _{N}=N\omega _{N-1}^{\frac {1}{N-1}}$) or ($\alpha =\alpha _{N};$ $b<N$).