A maximum principle

Question

Suppose that $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary $\partial\Omega$. Consider the elliptic boundary value problem for $\phi=\phi(x)$, $x\in\mathbb{R}^n$: \begin{equation}%%\label{eqn: cn cp phi system with steric effect stationary solns} \begin{cases} -\Delta \phi\geq \eta(\phi^{\ast})(\phi-\phi_0)\quad \text{in}\quad\hspace{2.5mm} \Omega,\\ \hspace{7mm}\phi=0 \hspace{28mm} \text{on}\quad \partial\Omega, \end{cases} \end{equation} where $\eta(\phi^{\ast})<0$, for some $\phi^{\ast}\in\mathbb{R}^1$ and $\phi_0\in\mathbb{R}^1$ is arbitrary. Does this problem enjoy the maximum principle? In other words, could we conclude that $\phi>0$ in $\Omega$?

Answer 1

It depends on the sign of $\phi_0$. You have $$ -\Delta \phi -\eta(\phi^*)\phi \ge -\eta(\phi^*)\phi_0. $$ If in addition $\phi_0$ is positive, then $\phi$ must be positive as well.