论文标题
涉及n-finsler-laplacian和l^p Norm的改进的Trudinger-Moser不平等现象
An improved Trudinger-Moser inequality involving N-Finsler-Laplacian and L^p norm
论文作者
论文摘要
假设$ f：\ mathbb {r}^{n} \ rightarrow [0， +\ infty）$是$ c^{2}类的凸函数（\ m athbb {r}^{n} {n} \ backslash \ backslash \ {0 \}） $γ_1= \ inf \ limits _ { u \ | _p^n}，$并定义标准$ $ \ | | | _ {n，f，γ，p} = \ bigG（\int_Ωf^{n}（\ nabla u）dx-u）dx-γ\ \ \ | | _p | _p | _p^n \ bigg \ bigG） \ Mathbb {r}^{n}（n \ geq 2）$是一个平稳的界面域。然后，对于$ p> 1 $和$ 0 \leqγ<γ_1$，我们有$$ \ sup_ {u \ in w^{1，n} _ {0}（ω），\ | u \ | _ { 1} \int_ΩE^{λ| u |^{\ frac {n} {n-1}}} dx <+\ infty，$ $ whene $ 0<λ\leqλ_{n} = n} = n} = n^{\ frac {\ frac {n} κ_{n}^{\ frac {1} {n-1}} $和$κ_{n} $是单位wulff球的体积。此外，通过使用爆破分析和容量技术，我们证明可以为任何$ 0 \leqγ<γ_1$获得上限。
Suppose $F: \mathbb{R}^{N} \rightarrow [0, +\infty)$ be a convex function of class $C^{2}(\mathbb{R}^{N} \backslash \{0\})$ which is even and positively homogeneous of degree 1. We denote $γ_1=\inf\limits_{u\in W^{1, N}_{0}(Ω)\backslash \{0\}}\frac{\int_ΩF^{N}(\nabla u)dx}{\| u\|_p^N},$ and define the norm $\|u\|_{N,F,γ, p}=\bigg(\int_ΩF^{N}(\nabla u)dx-γ\| u\|_p^N\bigg)^{\frac{1}{N}}.$ Let $Ω\subset \mathbb{R}^{N}(N\geq 2)$ be a smooth bounded domain. Then for $p> 1$ and $0\leq γ<γ_1$, we have $$ \sup_{u\in W^{1, N}_{0}(Ω), \|u\|_{N,F,γ, p}\leq 1}\int_Ωe^{λ|u|^{\frac{N}{N-1}}}dx<+\infty, $$ where $0<λ\leq λ_{N}=N^{\frac{N}{N-1}} κ_{N}^{\frac{1}{N-1}}$ and $κ_{N}$ 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 $0 \leqγ<γ_1$.
