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 \(W^{1, N}({\mathbb {R}}^N)\).

在本文中，假设\（F：{\ mathbb {R}} ^ {N} \ rightarrow [0，+ \ infty} \）是类\（C ^ {2}（{\ mathbb {R }} ^ {N} \反斜杠\ {0 \}）\），它的阶数为1且是正均匀的。首先，我们得出各向异性的Trudinger-Moser不等式，其中\（{\ mathbb {R}} ^ N的精确增长\），即对于任何\（b> 0 \），都有一个常量\（C_ {N，b}> 0 \）使得\（\ 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，\）其中\（\ 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}} \）和\（\ kappa _ {N} \ ）是\（{\ mathbb {R}} ^ N \）中单位Wulff球的体积。此外，如果将幂\（\ frac {N} {N-1} \）替换为任何\（p <\ frac {N} {N-1} \），则该不等式将失败。其次，我们计算超数的精确值，并给出关于不存在和最大化器存在的一些结果。最后，我们证明具有正增长的各向异性Trudinger-Moser不等式暗示\（W ^ {1，N}（{\ mathbb {R}} ^ N）\）中的Trudinger-Moser不等式。