Special Session 101: 

Minimization problem related to the critical Hardy inequality with non-decreasing potential function

Megumi Sano
Tokyo Institute of Technology
Japan
Co-Author(s):    
Abstract:
Let $a, q, \beta >1$, and $N \geq 2$. We consider the minimization problem associated with the optimal constant of the generalized critical Hardy inequalities as follows: $$ G_a := \inf_{0 \neq u \in W_0^{1,N}(B_1)} \frac{\int_{B_1} | \nabla u |^N dx}{\left( \int_{B_1} |u|^q f_a (x) dx \right)^{\frac{N}{q}} }, \,\text{where} \, f_a(x):= \frac{1}{|x|^N (\log \frac{a}{|x|})^{\beta}}. $$ In this talk, we show some results concerning the attainability of $G_a$ when $\beta=\frac{N-1}{N}q +1$ and $q>N$. Note that the potential function $f_a$ is not radially decreasing for $a$ close to 1. This is an open problem mentioned by T.Horiuchi in 2016.