On certain variant of strongly nonlinear interpolation inequality in dimension n

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subseteq {\bf R}^n$ and $n\ge 2$, $u:\Omega\rightarrow {\bf R}$ is in certain subset in second order Sobolev space $W^{2,1}_{loc}(\Omega)$, $\nabla^{(2)} u$ is the Hessian matrix of $u$, ${\cal T}_{h,C}(u)$ is certain transformation of the continuous function $h(\cdot)$. Such inequality is the generalization of similar inequality holding in one dimension, obtained earlier by second author and Peszek.