I want to get a estimation like $$ \left(\int_{\mathbb R^n} \nabla u_0\cdot \nabla u\right)^2 \ge C $$ where $C$ is a positive constant, and $u_0\in C^2(\mathbb R^n), u_0 > 0, |D^\alpha u_0|\le Ce^{-|x|},~~|\alpha|\le 2$. And $u\in H^1(\mathbb R^n), \|u\|_{L^2}=1$.
In fact, I think this estimation is wrong, But I am not sure. So, ask here.