Maximum principle for elliptic equation in exterior domain

Question

I have a question on maximum principle for elliptic equation in exterior domain.

Suppose that $u$ is infinitely differentiable in $\mathbb{R}^n$ and bounded in $\mathbb{R}^n$. I want to prove that if $u$ solves $$ a^{ij}D_{ij}u+b^i D_i u +cu-\lambda u=0\quad \text{in } \mathbb{R}^n\setminus B_1$$ where $a^{ij}$ is uniformly elliptic, $b^i$, $c$ are bounded and smooth with $c\leq 0$, and $\lambda>0$, then $$ |u(x)|\leq \max_{|y|=2} |u(y)|\quad \text{for all } |x|\geq 2. $$

If we assume some suitable decay condition on $u$, e.g., $\lim_{|x|\rightarrow \infty} u(x)=0$, then I can prove the desired assertion. My question is that can I prove it without imposing such decay condition? The assertion is not obvious to me if we replace our elliptic equation by $$ \triangle u -\lambda u=0\quad \text{in } \mathbb{R}^n \setminus B_1$$ although it is well-known that it possesses the maximum principle in the whole space.

My question is motivated by a recent paper of Krylov (https://arxiv.org/pdf/2004.01778.pdf) in page 14.

Also, I want to find any reference related to the maximum principle in the exterior domain.

Answer 1

Today, I realized that the proof is the direct corollary of the following lemma with the combination of the usual maximum principle.

For simplicity, we assume that $$ |a^{ij}|+|b^i|\leq K_1 $$ for some constant $K_1$.

Lemma. Let $\Omega=\mathbb{R}^n\setminus B_r$, and let $u$ be a continuous function bounded from above in $\overline{\Omega}$. Assume that $u$ is twice continuously differentiable and $\mathcal{L}u \geq 0$ in $\Omega$ and $\mathcal{L}1 \leq -1$. Here $$ \mathcal{L}u=a^{ij}D_{ij}u +b^i D_i u +cu.$$ Then there exists $\delta>0$ such that $$ u(x) \leq e^{\delta(r-|x|)} \max_{\partial \Omega} u^+. $$

This lemma is the Lemma 11.5.3 in the book of Krylov "Lectures on elliptic and parabolic equations in Sobolev spaces.