Extremal functions of generalized critical Hardy inequalities

TL;DR

This paper investigates the existence and properties of extremal functions for generalized critical Hardy inequalities, addressing an open problem and exploring how domain shape influences minimizers, including symmetry breaking for large parameters.

Contribution

It provides the first solutions to an open problem on Hardy inequalities in specific domains and analyzes how domain shape affects minimizers, including symmetry breaking phenomena.

Findings

Existence and non-existence of minimizers depending on domain shape.

Solution to an open problem for the ball domain.

Symmetry breaking of minimizers for large parameters.

Abstract

In this paper, we show the existence and non-existence of minimizers of the following minimization problems which include an open problem mentioned by Horiuchi and Kumlin in 2012: \begin{align*} G_a := \inf_{u \in W_0^{1,N}(\Omega ) \setminus \{ 0\} } \dfrac{\int_{\Omega} |\nabla u |^{N} \,dx}{\left( \int_{\Omega} |u|^{q} f_{a,\beta}(x) dx \right)^{\frac{N}{q}}}, \,\text{where} \,\,f_{a, \beta}(x):=|x|^{-N}\left( \log \frac{aR}{|x|} \right)^{-\beta}. \end{align*} First, we give an answer to the open problem when $\Omega =B_R(0)$. Next, we investigate the minimization problems on general bounded domains. In this case, the results depend on the shape of the domain $\Omega$. Finally, symmetry breaking property of the minimizers is proved for sufficiently large $\beta$.