For $\alpha \in [0, 1/2)$, let $h_\alpha: \mathbb{R}^2 \rightarrow [1, \infty)$ be defined by $x \rightarrow (1 + |x|)^\alpha$. I am interested in finding a $C^3(\mathbb{R}^2)$ function $\psi: \mathbb{R}^2 \rightarrow \mathbb{R}$ so that, for some value of $\alpha$, \begin{align*} \left\|\frac{\nabla \psi}{h_\alpha}\right\|_{L^\infty(\mathbb{R}^2)} &< \infty, \\ \left\|h_\alpha\nabla\Delta \psi\right\|_{L^\infty(\mathbb{R}^2)} &< \infty, \end{align*} and $$\left\|\Delta \psi\right\|_{L^\infty(B_R(0))} \rightarrow \infty \text{ as } R \rightarrow \infty,$$ where $B_R(0)$ is the ball of radius $R$ centered at the origin.
Can you either
- Demonstrate that some function $\psi$ exists that satisfies the above properties for some $\alpha \in [0, 1/2)$, or
- Prove no function $\psi$ exists satisfying these properties for any $\alpha \in [0, 1/2)$?