6 problems

Let $s\in(0,1)$ and let $n$ satisfy … so that $n\leq 3$. Let $\lambda$ be the parameter in the best constant $\mathcal S_s^{\lambda,{2^*_s}}(\mathbb R^n_+)$, and let…

Let $M_a(q)$ denote the Maz'ya constant associated with the purely cylindrical weighted minimization problem, and let $a$ and $q$ be as in that problem. A minimizer is radially dec…

Let $N\geq 3$, let $s\in(0,2)$, and let $U$ be the unique solution defined in the paper, associated with the Hardy–Sobolev problem on the half-space $\mathbb{R}^N_+$. A critical po…

Let $\Omega$ be a smooth bounded convex domain, and let $\mu_p(\Omega)$ denote the best constant in the Hardy–Sobolev inequality considered in the paper. Non-attainability conjectu…

Fractional Hardy–Sobolev optimizer conjecture. Up to constant multiples, rescaling, and, when $\alpha=0$, translations, every Aubin–Talenti function is

Let $s\in(0,1)$, let $n$ be the dimension, let $\lambda$ be in the range considered for $\mathcal S_s^{\lambda,{2^*_s}}(\mathbb R^n_+)$, and let…