Extremal functions for the anisotropic Sobolev inequalities

The existence of multiple nonnegative solutions to the anisotropic critical problem [GRAPHICS] 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-infinity(R-N) and is located outside of a ball of radius tau > 0 in L-P* (R-N). (c) 2006 Elsevier Masson SAS. All rights reserved.