An Improved Trudinger–Moser Inequality Involving N-Finsler–Laplacian and Lp Norm

Let $F: \mathbb {R}^{N} \rightarrow [0, +\infty )$ be a convex function of class $C^{2}(\mathbb \backslash \{0\})$ , which is even and positively homogeneous degree 1. ${\Omega }\subset {R}^{N}(N\geq 2)$ smooth bounded domain, we denote $\gamma _{1}=\inf \limits _{u\in W^{1, N}_{0}({\Omega })\backslash \{0\}}\frac {{\int }_{\Omega }F^{N}(\nabla u)dx}{\| u\|_{p}^{N}}$ define $\|u\|_{N,F,\gamma p}=\left ({\int u)dx-\gamma \| u\|_{p}^{N}\right )^{\frac {1}{N}}.$ Then for p > 1 0 ≤ γ < γ1, have $$ \sup_{u\in N}_{0}({\Omega}), \|u\|_{N,F,\gamma, p}\leq 1}{\int}_{\Omega}e^{\lambda_{N} |u|^{\frac{N}{N-1}}}dx<+\infty, where $\lambda _{N}=N^{\frac {N}{N-1}} \kappa _{N}^{\frac {1}{N-1}}$ κN the volume unit Wulff ball in $\mathbb {R}^{N}$ . Moreover, by using blow-up analysis capacity technique, prove that supremum can attained any γ1.