Set $\Omega$ an unbounded open set in $\mathbb{R^n}$, $L$ is the elliptic operator $Lu=-a_{ij}D_{ij}u+b_i D_i u+cu$ which satisfies the uniform elliptic condition. Also $c \ge 0$, $u \in C^2(\Omega) \bigcap C(\overline{\Omega})$ is the solution of the boundary condition problem: $$(i)Lu=f, x\in\Omega$$
$$(ii)u=\phi,x \in \partial \Omega$$
$$(iii)\lim_{x \to \infty,x \in\Omega} u(x) =0$$
Here we consider $a_{ij},b_{i},c,f,\phi$ is smooth enough and $b_i,f,\phi \in L^{\infty}(\Omega)$.
Prove that there is a constant C，independent of $u,\phi,f$ such that $$ \sup_{\Omega}|u| \le \sup_{\partial \Omega}|\phi| + C\sup_{\Omega}|f| $$
Itˊs easy to make the conclusion when $\Omega$ is bounded or $\Omega$ is replaced by $\Bbb{R^n}-U $, where $U$ is a bounded open region(exterion Dirichlet problem). But since $\Omega$ is an unbounded open set,itˊs hard to use similar method to solve it. How to make the first step to this problem?