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

Abstract

Suppose $F:{\mathbb{R}}^N\to [0,+\infty )$ be a convex function of class $C^2({\mathbb{R}}^N\backslash \{0\})$ which is even and positively homogeneous of degree 1. We denote ${\hat{I}}_1^3=\underset{u\in W_0^{1,N}(\hat{I}\text{c◯})\backslash \{0\}}{\inf }\frac{{\int }_{\hat{I}}\text{c◯}{F}^N(\nabla u)dx}{\Vert u{\Vert }_p^N},$ and define the norm $\Vert u{\Vert }_{N,F,{\hat{I}}^3,p}=({\int }_{\hat{I}}\text{c◯}{F}^N(\nabla u)dx-{\hat{I}}^3\Vert u{\Vert }_p^N{)}^{\frac{1}{N}}.$ Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^N(N\ge 2)$ be a smooth bounded domain. Then for $p>1$ and $0\le {\hat{I}}^3<{\hat{I}}_1^3$, we have $$\underset{u\in W_0^{1,N}(\hat{I}\text{c◯}),\Vert u{\Vert }_{N,F,{\hat{I}}^3,p}\le 1}{\sup }{\int }_{\hat{I}}\text{c◯}{e}^{\hat{I}»|u{|}^{\frac{N}{N-1}}}dx<+\infty ,$$ where $0<\hat{I}»\le \hat{I}{»}_N=N^{\frac{N}{N-1}}\hat{I}{º}_N^{\frac{1}{N-1}}$ and $\hat{I}{º}_N$ 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 $0\le {\hat{I}}^3<{\hat{I}}_1^3$.

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