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

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 $0 \leq\gamma <\gamma_1$.
Comment: 33 Pages