Extremal functions for the anisotropic Sobolev inequalities

Abstract

The existence of multiple nonnegative solutions to the anisotropic critical problem−∑i=1N∂∂xi(|∂u∂xi|pi−2∂u∂xi)=|u|p∗−2uinRN 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. Furthermore, we show that the set of nontrival solutions S is included in L∞(RN) and is located outside of a ball of radius τ>0 in Lp∗(RN).