Anisotropic Trudinger–Moser inequalities associated with the exact growth in ${\mathbb {R}}^N$ and its maximizers

Abstract

In this paper, 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. Firstly, we derive anisotropic Trudinger–Moser inequality with exact growth in ${\mathbb {R}}^N$, i.e., for any $b>0$, there exists a constant $C_{N, b}>0$ such that $\int _{{\mathbb {R}}^{N}}\frac{\varPhi _N(\lambda |u|^{\frac{N}{N-1}})}{1+b|u|^{\frac{N}{N-1}}}dx \le C_{N, b}\Vert u\Vert _N^N, \quad \forall u\in W^{1, N}({\mathbb {R}}^{N}) \quad \text {with} \quad \int _{{\mathbb {R}}^N}F^{N}(\nabla u)dx \le 1,$ where $\varPhi _N(t):=e^t-\sum _{k=0}^{N-2}\frac{t^k}{k!}$, $\lambda \le \lambda _{N}=N^{\frac{N}{N-1}} \kappa _{N}^{\frac{1}{N-1}}$ and $\kappa _{N}$ is the volume of a unit Wulff ball in ${\mathbb {R}}^N$. Moreover, this inequality fails if the power $\frac{N}{N-1}$ is replaced by any $p<\frac{N}{N-1}$. Secondly, we calculate the exact values of the supremums and give some results about nonexistence and existence of maximizers. Finally, we prove that anisotropic Trudinger–Moser inequality with the exact growth implies Trudinger–Moser inequality in $W^{1, N}({\mathbb {R}}^N)$.