# real analysis – Nonnegative super(sub)-harmonic functions for an elliptic operator

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.