In this paper, we investigate sharp singular Trudinger-Moser inequalities involving the anisotropic Dirichlet norm $\left(\int_{ \Omega}F^{ N}(\nabla u)\;\mathrm{ d}x\right)^{ \frac{ 1}{ N}}$ in the Sobolev-type space $D^{ N,q}(\mathbb{ R}^{ N})$, $q\geq 1$, here $F:\mathbb{ R}^{ N}\rightarrow[0,+\infty)$ is a convex function of class $C^{ 2}(\mathbb{ R}^{ N}\setminus\{ 0\})$, which is even and positively homogeneous of degree 1, its polar $F^{ 0}$ represents a Finsler metric on $\mathbb{ R}^{ N}$. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger-Moser inequalities and discuss their sharpness under several different situations, including the case $\|F(\nabla u)\|_{ N}\leq 1$, the case $\|F(\nabla u)\|_{ N}^{ a}+\|u\|_{ q}^{ b}\leq 1$, and whether they are associated with exact growth.
Categories: math.FA