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=(1-|x|^2)Delta f-c((x-theta),nabla f),

end{align*}

where $theta in U$ and $c$ is a positive constant.

**My question.** **Can we find a smooth and nonnegative function $fcolon U to mathbb{R}$ 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 range of $c$ may be limited.

If $theta=0$, we can find such a function. Indeed, if we set $f=1-(1-|x|^2)^{alpha}$, $alpha in (0,1)$, 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 (there may be something simpler than this). If $theta neq 0$, however, I could not find a function satisfying the above conditions.

If you find one, please let me know.