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Consider the eigenfunction $\varphi_R>0$ $$L\varphi_R=\lambda_R\varphi_R, \ \ \ in \ \ B_R,$$ and $$\varphi_R=0, \ \ \ in \ \ \partial B_R,$$ where $L$ is a elliptic operator and $\lambda_R$ is the corresponding principal eigenvalue. We can normalize $\varphi_R$ such that $\varphi_R(0)=1$. Then, for the "Harnack inequality" in the ball $B_{2R}$, exists a constant $\delta_R>0$ such that $$\delta_R\leq\varphi_R\leq\delta_R^{-1}, \ \ \ in \ \ \overline{B_{3R/2}}.$$

Where can I find this Harnack inequality?

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I think some of your $R$s and $2R$s are mixed up... – Nate Eldredge Nov 22 '12 at 14:18
What this means? – José Carlos Nov 22 '12 at 14:21
How can you expect $\varphi_R$ to satisfy a Harnack inequality on $B_{3R/2}$ if it's only a solution in $B_R$? – Nate Eldredge Nov 22 '12 at 14:24
After you fix the ratio of the ball you can try: Corollary 9.25, page 250 from the book of Gilbard Trudinger – Tomás Nov 22 '12 at 14:28
Nate, i think that $\varphi_R$ is defined in a domain bigger than $B_R$. But the first equality is only satisfies in the ball. – José Carlos Nov 22 '12 at 14:31

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