Inserted: 29 jul 2003
Last Updated: 19 jan 2005
Journal: Ann. Inst. H. Poincaré
We consider a new class of quasilinear elliptic equations with a power-like reaction term: the differential operator weights partial derivatives with different powers, so that the underlying functional-analytic framework involves anisotropic Sobolev spaces. Critical exponents for embeddings of these spaces into $L^q$ have two distinct expressions according to whether the anisotropy is ``concentrated'' or ``spread out''. Existence results in the subcritical case are affected by this dichotomy. On the other hand, nonexistence results are obtained in the at least critical case in domains with a geometric property which modifies the standard notion of starshapedness.