I am looking for a super(sub) harmonic function for an elliptic operator.
Let $n$ be a positive integer. We denote by $(\cdot,\cdot)$ and $|\cdot|$ the standard inner product and norm on $\mathbb{R}^n$, respectively. We denote by $U \subset \mathbb{R}^n$ the open unit ball centered at the origin. The elliptic operator $\mathcal{L}$ is defined as follows: \begin{align*} \mathcal{L}f(x)=(1-|x|^2)\Delta f-c((x-\theta),\nabla f),\quad x \in U, \end{align*} where $\theta \in U$ and $c$ is a positive constant.
My question. Can we find a smooth and nonnegative function $f\colon U \to \mathbb{R}$ and $\varepsilon>0$ such that $\mathcal{L}f \ge \varepsilon$ on $U $ ? Needless to say, the function $f$ may depend on $\theta$. If necessarily, the ranges of $c$ and $|\theta|$ may be limited. If we find such a function, in a sense, we can understand the boundary behavior of the diffusion process associated with $\mathcal{L}$.
If $\theta=0$, we can find such a function. Indeed, if we set $f=\alpha^{-1}\{1-(1-|x|^2)^{\alpha}\}$, $\alpha \in (0,1)$ (there may be something simpler than this), we obtain that \begin{equation} \mathcal{L}f=4\{(1-c/2)-\alpha \}|x|^2(1-|x|^2)^{\alpha-1}+2n(1-|x|^2)^{\alpha}. \end{equation} Therefore, if $c<2$ and $\alpha \in (0,1-c/2)$, we find that $f$ possesses the desired property (in fact, $c=2$ is a border in a sense).
If $\theta \neq 0$, however, I could not find a function satisfying the above conditions.
If you find one, please let me know.
