Missing terms in generalized Hardy’s inequalities and its applications

Abstract

In this article we shall investigate the Hardy inequalities and improve them by finding out missing terms. Although the missing terms for the higher order Hardy inequality can not be determined in a unique way, we shall give a canonical form of the remainder. As a direct application we shall study blow-up solutions of a semilinear elliptic boundary value problem and give some lower estimate of the first eigenvalue of the linearized operator. We also improve the weighted Hardy inequalities, which will be fundamental to study singular solutions of quasilinear elliptic equations.