On Hardy-Rellich type inequalities in \(\mathbb R^N\)
The following result is proved. Theorem. For all \(a,b\in\mathbb R\) and \(u\in C_0^{\infty}(\mathbb R^N\setminus\{0\})\) one has \[ \biggl[(n-2-\gamma)\int_{\mathbb R^N}\frac{|\nabla u|^2}{|x|^{\gamma}}\,dx+ 2\gamma\int_{\mathbb R^N}\frac{(x\cdot\nabla u)^2}{|x|^{\gamma+2}}\,dx\biggr]^2\leq 4\biggl(\int_{\mathbb R^N}\frac{|\Delta u|^2}{|x|^{2b}}\,dx\biggr)\biggl(\int_{\mathbb R^N}\frac{|\nabla u|^2}{|x|^{2a}}\,dx\biggr) \] where \(\gamma:=a+b+1\). In addition, if \(\gamma\leq n-2\), then \[ \biggl|\frac{N+a+b-1}{2}\biggr|\int_{\mathbb R^N}\frac{|\nabla u|^2}{|x|^{a+b+1}}\,dx\leq \biggl(\int_{\mathbb R^N}\frac{|\Delta u|^2}{|x|^{2b}}\,dx\biggr)^{1/2} \biggl(\int_{\mathbb R^N}\frac{|\nabla u|^2}{|x|^{2a}} \,dx\biggr)^{1/2}, \] where the constant \(|\frac{N+a+b-1}{2}|\) is sharp.
scientific article; zbMATH DE number 5563917