Extremal functions for the anisotropic Sobolev inequalities

TL;DR

This paper proves the existence of multiple solutions to an anisotropic critical PDE, identifying extremal functions for the best Sobolev constant using adapted concentration-compactness techniques.

Contribution

It introduces an adaptation of Lions' concentration-compactness lemma for anisotropic operators and establishes properties of the solution set.

Findings

Existence of multiple nonnegative solutions to the anisotropic critical problem.

Solutions are extremal functions of a best Sobolev constant.

Solution set is bounded in L^ abla and outside a positive radius in L^{p^*}.

Abstract

The existence of multiple nonnegative solutions to the anisotropic critical problem - \sum_{i=1}^{N} \frac{\partial}{\partial x_i} (| \frac{\partial u}{\partial x_i} |^{p_i-2} \frac{\partial u}{\partial x_i}) = |u|^{p^*-2} u {in} \mathbb{R}^N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions $\calS$ is included in $L^\infty(\R^N)$ and is located outside of a ball of radius $\tau >0$ in $L^{p^*}(\R^N)$.