OPTIMAL HARDY WEIGHTED INEQUALITIES AND MULTIPLE POSITIVE SOLUTIONS FOR QUASI-LINEAR ELLIPTIC EQUATIONS

Abstract

We show the role of some Hardy inequalities in the blow-up phenomena of the very weak solution of a linear equation in the sense of Brezis. We find some new Hardy inequalities related to some extended Sobolev spaces, Sobolev–Hardy spaces andSobolev– Zygmund spaces, or other non-standard weighted spaces. The method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions. We study the multiplicity results of positive solutions for a semi-linear elliptic system involving both concave–convex and critical growth terms. We also study the multiplicity results of positive solutions for a class of quasi-linear elliptic equations involving critical Sobolev exponent.