A critical Hardy-Rellich inequality

Abstract

In this work, we prove a critical version of a Hardy-Rellich type inequality. We show that for $N\geq 1$ and for $-N 0$ such that \[ \int_{\mathbb R^N}\left|\nabla\left(\frac{u(x)}{|x|}\right)\right|^N|x|^a\,\mathrm{d}x\leq C_{N,a}\int_{\mathbb R^N}\left|\Delta u(x)\right|^N |x|^a\,\mathrm{d}x, \] for any $u\in C^\infty_c(\mathbb R^N\setminus\left\{0\right\})$.