| 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. |
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