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Background: I am aware of the Harnack's Inequality for linear elliptic equations.

My questions are:

(a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form $$Lu = |u|^pu ?$$ where $L$ is elliptic? Edit: This I understand now. Please look at (b) and (c). I am leaving this for the sake of "completeness"...

(b) How about the same question (I am actually more interested in this) when $L$ is hypoelliptic?

(c) Suppose I have a non-negative solution to the equation $Lu = |u|^pu $, with $L$ being elliptic/hypoelliptic. I want to show that the solution is actually positive. What other generic method is there to proceed other than Harnack's Inequality? I realise this last question is kind of vague, but I hope the intent behind the question is clear.

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  • $\begingroup$ For (c) you don't need a Harnack inequality, you can use the strict maximum principle $\endgroup$
    – Robert Haslhofer
    Apr 13, 2013 at 1:21
  • $\begingroup$ @Robert Haslhofer Is there a maximum principle for hypoelliptic operators? Can you please point me to a source? Thanks! $\endgroup$
    – grateful
    Apr 13, 2013 at 2:00
  • $\begingroup$ @grateful: No there is no maximum principle in general. Maximum principle is valid only for second order operators. $\endgroup$
    – timur
    Dec 21, 2013 at 16:39

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