Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations

TL;DR

This paper studies the long-term behavior of solutions to anisotropic equations with critical growth, establishing decay rates and revealing a vanishing phenomenon linked to the exponents' critical values.

Contribution

It provides new decay estimates and uncovers a vanishing phenomenon for solutions of anisotropic equations with critical Sobolev growth.

Findings

Decay estimates for solutions and derivatives

Identification of a vanishing phenomenon when exponents exceed a critical value

Connection to extremal functions for anisotropic Sobolev inequalities

Abstract

We investigate the asymptotic behavior of solutions of anisotropic equations of the form $-\sum_{i=1}^n\partial_{x_i}(\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u)=f(x,u)$ 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 exponents $p_i$ exceeds a critical value.