Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities

Let $N\ge 2$, $q>1$, and $0<\beta <N$, we study extremals for the critical singular anisotropic Moser–Trudinger functional under $$\parallel F(\mathrm{\nabla }u){\parallel }_N^a+\parallel u{\parallel }_q^b\le 1,a>0,0<b\le N\mathrm{.}$$ For $b<N$, the supremum is attained. At $b=N$, put $$q_{-}:=\frac{N^2(N-2)}{(N-1)(N-\beta )},q_{+}:=\frac{N^2}{N-\beta }\mathrm{.}$$ For $q<q_{-}$, the supremum is attained for every $a>0$. For $q_{-}\le q<q_{+}$, there exists $a_c\in (N,\mathrm{\infty }]$ such that the supremum is attained for $0<a<a_c$ and is not attained for $a>a_c$ if $a_c<\mathrm{\infty }$. The argument does not determine attainment at $a=a_c$. Radial blow-up analysis gives a concentration bound in terms of a nonlinear Green function. The first nonzero Taylor term determines the strict comparison, an exact Euclidean reduction shows that the threshold is independent of $F$, a separate supplement gives an alternative proof for $q=N$ using the classical Green function and compact Taylor terms.