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

Abstract

In this paper, suppose $F:{\mathbb{R}}^N\to [0,+\infty )$ 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 $\hat{I}\text{c◯}\subset {\mathbb{R}}^N(N\ge 2)$ be a smooth bounded domain. $\{u_n\}\subset W_0^{1,N}(\hat{I}\text{c◯})$ be a sequence such that anisotropic Dirichlet norm${\int }_{\hat{I}}\text{c◯}{F}^N(\nabla u_n)dx=1$, $u_n\rightharpoonup u\not\equiv 0$ weakly in $W_0^{1,N}(\hat{I}\text{c◯})$. Then for any $0<p<p_N(u):=(1-{\int }_{\hat{I}}\text{c◯}{F}^N(\nabla u)dx{)}^{-\frac{1}{N-1}},$ we have $${\int }_{\hat{I}}\text{c◯}\frac{e^{\hat{I}{»}_N(1-\frac{\hat{I}}{²}N)p|u_n{|}^{\frac{N}{N-1}}}}{F^o(x{)}^2}dx<+\infty ,$$ where $0\le {\hat{I}}^2<N$, $\hat{I}{»}_N=N^{\frac{N}{N-1}}\hat{I}{º}_N^{\frac{1}{N-1}}$ and $\hat{I}{º}_N$ 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 ${\mathbb{R}}^N$.

20 Pages