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.

  • $\begingroup$ You need growth conditions on norms of $f$ for large $s$ to make such a conclusion. Just look at a proof of a Harnack inequality for general elliptic pdi and follow it through for your pdi and refine it as much as possible. See for example "The Maximum Principle" by Pucci and Serrin for a general self contained proof of an associated Harnack inequality. $\endgroup$ – JCM Jan 2 '19 at 22:12

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