  We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq
C(n,p)\int_{\Omega} \left( \sqrt{ |\Delta u(x)||{\cal
T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subset \mathbf{R}^n$ is a
bounded Lipschitz domain, $u\in W^{2,1}_{loc}(\Omega)$ is postive and obeys
some additional assumptions, $\Delta u$ is the Laplace operator, ${\cal
T}_{h,C}(\cdot )$ is certain transformation of the continuous function
$h(\cdot)$. We also explain how to apply such inequality to deduce regularity
for solutions of nonlinear eigenvalue problems of elliptic type for degenerated
PDEs, with the illustration within the model of electrostatic micromechanical
systems (MEMS).
