Estimation for a maximum principle

Question

Suppose $-\Delta u(x) + u(x)c(x) \leq 0$, for all $x \in \Omega$. Worth the inequality $$-\Delta v(x) - \dfrac{2}{\phi(x)}\nabla\phi(x)\nabla v(x) + Kv(x) \leq 0,$$ where $K:= \lambda + c(x) > 0$, $v(x) := u(x)/\phi(x)$ and $-\Delta \phi = \lambda\phi$, for each $x \in \Omega$?