We assume $u\ge0$ satisfies the following elliptic equation with parameter $s>0$: $$ -\Delta u(x;s)+s^2u(x;s)=f(x;s)\le0\mbox{ in $B_1\subset\mathbb{R}^n$.} $$ Here $n\ge1$, $B_1$ is the unit ball in $\mathbb{R}^n$ and $f(\cdot;s)\in L^\infty(B_1)$ for any $s>0$.
My question: can we have the Harnack inequality as follows: $$ \sup_{B_{1/2}} u(x:s) \le C(s)\left(\inf_{B_{1/2}}u(x;s) + \|f\|_{L^\infty(B_1)}\right), $$ Moreover, what is the dependency of the constant $C$ on $s$ for $s>0$ being sufficiently large, for example, can we have $$ C(s)\le \exp(Cs),\mbox{ $s>0$ large enough.} $$
If it is true, then how to prove it? Thank you very much.
