Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator: existence of extremal functions, breaking positivity and breaking symmetry

TL;DR

This paper explores advanced inequalities related to the weighted biharmonic operator on cones, focusing on the existence, sign-changing behavior, and symmetry-breaking of extremal functions under various boundary conditions.

Contribution

It introduces new results on the existence and qualitative properties of extremal functions for Caffarelli-Kohn-Nirenberg type inequalities involving the weighted biharmonic operator.

Findings

Extremal functions can change sign in certain cases.

Symmetry-breaking phenomena occur in the whole space setting.

Existence of extremal functions is established under various boundary conditions.

Abstract

We investigate Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator on cones, both under Navier and Dirichlet boundary conditions. Moreover, we study existence and qualitative properties of extremal functions. In particular, we show that in some cases extremal functions do change sign; when the domain is the whole space, we prove some breaking symmetry phenomena.