---
abstract: 'We investigate the asymptotic behavior of solutions of anisotropic equations of the form $-\sum_{i=1}^n\partial_{x_i}\big(\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u\big)=f{\left(}x,u{\right)}$ in ${\mathbb{R}}^n$, where $p_i>1$ for all $i=1,\dotsc,n$ and $f$ is a Caratheodory function with critical Sobolev growth. This problem arises in particular from the study of extremal functions for a class of anisotropic Sobolev inequalities. We establish decay estimates for the solutions and their derivatives, and we bring to light a vanishing phenomenon which occurs when the maximum value of the $p_i$ exceeds a critical value.'
address: 'Jérôme Vétois, Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06108 Nice, France.'
author:
- Jérôme Vétois
date: 'October 2, 2014. [*Final version:*]{} July 22, 2015.'
title: Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations
---

[^1]

Introduction and main results {#Sec1}
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We let $n\ge2$ and $\operatorname{\overrightarrow{p}}={\left(}p_1,\dotsc,p_n{\right)}$ be such that $p_i>1$ for all $i=1,\dotsc,n$ and $\sum_{i=1}^n1/p_i>1$. In this paper, we are interested in the solutions of problems of the form $$\label{Eq1}
\left\{\begin{aligned}&-\Delta_{\operatorname{\overrightarrow{p}}} u=f{\left(}x,u{\right)}\quad\text{in }{\mathbb{R}}^n,\\
&u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)},
\end{aligned}\right.$$ where $\Delta_{\operatorname{\overrightarrow{p}}} u:=\sum_{i=1}^n\partial_{x_i}{\left(}\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u{\right)}$ is the anisotropic Laplace operator, $D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$ is the completion of $C^\infty_c{\left(}{\mathbb{R}}^n{\right)}$ with respect to the norm $\left\|u\right\|_{D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}:=\sum_{i=1}^n{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{1/p_i}$, and $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ is a Caratheodory function such that $$\label{Eq2}
\left|f{\left(}x,s{\right)}\right|\le\Lambda\left|s\right|^{p^*-1}\quad\text{for all }s\in{\mathbb{R}}\text{ and a.e. 