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

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 $i=1,\dots ,n$ 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.

Final version to appear in Advances in Mathematics

References in corpus (3)

• Classification of positive ${\mathcal{D}}^{1,p}({\mathbb{R}}^N)$-solutions to the critical $p$-Laplace equation in ${\mathbb{R}}^N$
• Extremal functions for the anisotropic Sobolev inequalities
• A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations

Cited by in corpus (1)

• Extremal functions for an embedding from some anisotropic space, and partial differential equation involving the "one Laplacian"