Elliptic Harnack inequality for 1D Schrodinger operator?

Question

For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is:

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There exist $C_{H} > 0$ and $\delta \in (0,1)$ such that for any $B(x,r)$ in $\mathbb{R}$, if $u \geq 0$ satisfies $Hu \equiv 0$ in $B(x,r)$, then

$$\sup_{B(x,\delta r)} u \leq C_{H} \inf_{B(x,\delta r)} u$$

where $C_{H}$ and $\delta$ are indepedent of $x$ and $r$.

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I see results such as the "strong Harnack inequality" in Aizenman and Simon's 1982 paper "Brownian Motion and Harnack Inequality for Schrödinger Operators". But their equivalent of the constant $C_{H}$ appears to depend on BOTH $x$ and $r$.

Answer 1

The constant $C_H$ must depend on $x$ and $r$. Consider for example $u = e^{\frac{1}{2}|x|^2}$, which satisfies $$\Delta u = (1+|x|^2)u.$$ Then $u$ has a minimum of $1$ at $0$, which clearly doesn't control the maximum of $u$ on $B_{\delta r}$ times any constant independent of $r$.

Furthermore, $C_H$ must depend on $x$ because if we fix $r$ then the ratio $$\frac{u(x+r)}{u(x-r)}$$ blows up as $x \rightarrow \infty$.

Answer 2

There is a nice presentation of Harnack inequalities for linear elliptic p.d.e in Protter and Weinberger "Maximum Principles in Differential Equations". Moreover, there are additional references to the nonlinear case in the bibliography, which are probably more helpful.