Sharp anisotropic singular Trudinger-Moser inequalities in the entire space (2305.12443v1)
Abstract: 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.