$\mathcal{C}^2_0(\Omega)$ denote the functions $2$ times differentiable in $\Omega \subset \mathbb{R}^n$ with compact support, i.e., which vanishes in some compact $K\subset \Omega.$
$B_r=B(0;r)$
If $v \in \mathcal{C}^2(B_1), \ w=|Dv|^2$ and $\varphi \in \mathcal{C}_0^2(B_1)$ is a cutoff nonegative function, then by the Hölder inequality we have
\begin{equation*} \begin{split} \Delta(\varphi w)+2\sum_{i=1}^nv_i(\varphi w)_i&=2\varphi \sum_{i=1}^n\sum_{j=1}^nv_{ij}^2+4\sum_{i=1}^n\sum_{j=1}^n\varphi_iv_jv_{ij}+2w\sum_{i=1}^n\varphi_iv_i+(\Delta\varphi)w\\ &\geq \varphi\sum_{i=1}^n\sum_{j=1}^n v_{ij}^2-2|D\varphi||Dv|^3-\left(|\Delta\varphi|+C\dfrac{|D\varphi|^2}{\varphi}\right)|Dv|^2, \end{split} \end{equation*} where $v_i=\dfrac{\partial v}{\partial x_i}, \ v_{ij}=\dfrac{\partial^2 v}{\partial x_ix_j} \ldots$ and choosing $\varphi$ such that $\frac{|D\varphi|^2}{\varphi}$ is bounded in $B_1.$
I haven't got to understand why this inequalitity happens, because I don't see how to apply the Hölder inequality here. Please, any hint or suggestion would be helpful.
