Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for all $x\in\overline\Omega$. Can will find $\delta>0$ such that $$f(u)=_{df} \int_\Omega u\frac{\Delta\Phi}{\Phi} + \Phi \frac{\Delta u}{u}\ge 0$$ for all $u\in B_\delta(\Phi)$, where $B_\delta(\Phi)=\{u\in S: \|u-\Phi\|_{C^1(\overline\Omega)}<\delta \}$?

I only know that $f(\Phi)=0$ and $f(\Phi+\epsilon)\ge 0$ if $\epsilon$ is small. Can any one prove or find a counterexample or suggest me some possible methods to approach it? Thanks....