Concentration-compactness principle of singular Trudinger-Moser inequality involving $N$-Finsler-Laplacian operator

In this paper, suppose $F:{\mathbb{R}}^N\to [0,+\mathrm{\infty })$ be a convex function of class $C^2({\mathbb{R}}^N\mathrm{\backslash }\{0\})$ which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger-Moser Inequalities involving $N$-Finsler–Laplacian operator. Let $\Omega\subset $\mathbb{R}$^{N}(N\geq 2)$beasmoothboundeddomain\mathrm{.}${u_n}\subset W_0^{1, N}(\Omega)$beasequencesuchthatanisotropicDirichletnorm$\int_{\Omega}F^N (\nabla u_n)dx=1$,$u_n \rightharpoonup u \not \equiv 0$weaklyin$W_0^{1, N}(\Omega)$\mathrm{.}Thenforany$0 < p < p_N(u):=(1-\int_{\Omega}F^N (\nabla u)dx)^{-\frac{1}{N-1}},$wehave$$ \int_{\Omega}\frac{e^{\lambda_{N}(1-\frac{\beta}{N})p |u_n|^{\frac{N}{N-1}}}}{F^{o}(x)^{\beta}}dx<+\infty, $$where$0\leq\beta <N$,$\lambda_{N}=N^{\frac{N}{N-1}} \kappa_{N}^{\frac{1}{N-1}}$and$\kappa_{N}$ is the volume of a unit Wulff ball. This conclusion fails if $p\ge p_N(u)$. Furthermore, we also obtain the corresponding concentration-compactness principle in the entire Euclidean space ${\mathbb{R}}^N$.