SHARP SOBOLEV ASYMPTOTICS FOR CRITICAL ANISOTROPIC EQUATIONS ABDALLAH EL HAMIDI AND JEROME VETOIS Abstract. We investigate blow-up theory and prove sharp Sobolev asymptotics for a general class of anisotropic critical equations in bounded domains of the Euclidean space. 1. Introduction and statement of the results We consider in this paper critical anisotropic equations in bounded domains of the Euclidean space. Anisotropic operators appear in several places in the literature. Recent references can be found in physics [13, 17, 18, 23, 24], in biology [10, 11], and in image processing (see, for instance, the monograph by Weickert [50]). By definition, anisotropic operators involve directional derivatives with distinct weights. Given an open subset ? of Rn, n ≥ 2, and ??p = (p1, . . . , pn), we let D1, ??p (?) be the Sobolev space defined as the completion of the vector space of all smooth functions with compact support in ? with respect to the norm ?u?D1,??p (?) = n∑ i=1 ? ? ? ? ∂u ∂xi ? ? ? ? Lpi (?) . We let also p? be the corresponding critical exponent for the embeddings of the anisotropic Sobolev space D1, ??p (?) into Lebesgue spaces. We assume that the exponents pi satisfy n∑ i=1 1 pi > 1 and 1 < pi < n ∑n j=1 1 pj ? 1 for i = 1, .
- sobolev space
- properties hold
- ??p
- stable domains
- corresponding critical exponent
- p? ?
- critical anisotropic
- up