An improved Trudinger-Moser inequality involving N-Finsler-Laplacian and L^p norm

Abstract: 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 denote $\gamma_1=\inf\limits_{u\in W^{1, N}{0}(\Omega)\backslash {0}}\frac{\int{\Omega}F^{N}(\nabla u)dx}{| u|p^N},$ and define the norm $|u|{N,F,\gamma, p}=\bigg(\int_{\Omega}F^{N}(\nabla u)dx-\gamma| u|p^N\bigg)^{\frac{1}{N}}.$ Let $\Omega\subset \mathbb{R}^{N}(N\geq 2)$ be a smooth bounded domain. Then for $p> 1$ and $0\leq \gamma <\gamma_1$, we have $$ \sup{u\in W^{1, N}{0}(\Omega), |u|{N,F,\gamma, p}\leq 1}\int_{\Omega}e^{\lambda |u|^{\frac{N}{N-1}}}dx<+\infty, $$ where $0<\lambda \leq \lambda_{N}=N^{\frac{N}{N-1}} \kappa_{N}^{\frac{1}{N-1}}$ and $\kappa_{N}$ is the volume of a unit Wulff ball. Moreover, by using blow-up analysis and capacity technique, we prove that the supremum can be attained for any $0 \leq\gamma <\gamma_1$.