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.
