The quasi-linear Brezis-Nirenberg problem in low dimensions

Abstract

We discuss existence results for a quasi-linear elliptic equation of critical Sobolev growth [H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437--477; M. Guedda, L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), no. 8, 879--902] in the low-dimensional case, where the problem has a global character which is encoded in sign properties of the ``regular" part for the corresponding Green's function as in [O. Druet, Elliptic equations with critical Sobolev exponents in dimension 3, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 19 (2002), no. 2, 125--142; P. Esposito, On some conjectures proposed by Ha\"im Brezis, Nonlinear Anal. 54 (2004), no. 5, 751--759].