A critical Hardy-Rellich inequality

Abstract

In this work, we prove a critical version of a Hardy-Rellich type inequality. We show that for $N\ge 1$ and for $-N<a<1$, $a\ne 0$ there exists a constant $C_{N,a}>0$ such that \[ \int_{\mathbb R^N}\left|\nabla\left(\frac{u(x)}{|x|}\right)\right|^N|x|^a\,\mathrm{d}x\leq C_{N,a}\int_{\mathbb R^N}\left|Δu(x)\right|^N |x|^a\,\mathrm{d}x, \] for any $u\in C_c^{\infty }({\mathbb{R}}^N\setminus \left\{0\right\})$.