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

Abstract

In this paper, suppose $F: \mathbb{R}^{N} \rightarrow [0, +\infty)$ be a convex function of class $C^{2}(\mathbb{R}^{N} \backslash \{0\})$ 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 $\Omega\subset \mathbb{R}^{N}(N\geq 2)$ be a smooth bounded domain. $\{u_n\}\subset W_0^{1, N}(\Omega)$ be a sequence such that anisotropic Dirichlet norm$\int_{\Omega}F^N (\nabla u_n)dx=1$, $u_n \rightharpoonup u \not \equiv 0$ weakly in $W_0^{1, N}(\Omega)$. Then for any $0 < p < p_N(u):=(1-\int_{\Omega}F^N (\nabla u)dx)^{-\frac{1}{N-1}},$ we have $$ \int_{\Omega}\frac{e^{\lambda_{N}(1-\frac{\beta}{N})p |u_n|^{\frac{N}{N-1}}}}{F^{o}(x)^{\beta}}dx<+\infty, $$ 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 $p \geq p_N(u)$. Furthermore, we also obtain the corresponding concentration-compactness principle in the entire Euclidean space $\mathbb{R}^{N}$.