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We investigate the asymptotic behavior of solutions of anisotropic equationsof 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 aCaratheodory function with critical Sobolev growth. This problem arises inparticular from the study of extremal functions for a class of anisotropicSobolev inequalities. We establish decay estimates for the solutions and theirderivatives, and we bring to light a vanishing phenomenon which occurs when themaximum value of the exponents $p_i$ exceeds a critical value.

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