Sharp second order inequalities with distance function to the boundary and applications to a p-Biharmonic singular problem

TL;DR

This paper generalizes Hardy-Rellich inequalities in the L^p setting for domains with boundary singularities, providing sharp bounds and applications to singular p-Biharmonic problems using variational methods.

Contribution

It introduces new sharp inequalities involving the distance to the boundary and applies them to analyze existence of solutions for singular p-Biharmonic problems.

Findings

Established sharp Hardy-Rellich inequalities in L^p setting.

Derived bounds for the sharp constant depending on domain geometry.

Applied inequalities to prove existence and non-existence results for singular problems.

Abstract

In this paper, we prove generalizations to the L^p setting of the Hardy-Rellich inequalities on domains of R^N with singularity given by the distance function to the boundary. The inequalities we obtain are either sharp in bounded domains, where we provide concrete minimizing sequences, or give a new bound for the sharp constant, while also depending on the geometric properties of the domain and its boundary. We also give applications to the existence and non-existence of solutions for a singular problem using variational methods and a Pohozaev identity.