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

We investigate the asymptotic behavior of solutions of anisotropic equations of the form −∑i=1n∂xi(|∂xiu|pi−2∂xiu)=f(x,u) in RnRn, where pi>1pi>1 for all i=1,…,ni=1,…,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 pipi exceeds a critical value.