Harnack's inequality for general solutions with nonstandard growth

We prove Harnack's inequality for general solutions of elliptic equations idivA(x;u;ru) = B(x;u;ru); where A and B satisfy natural structural conditions with respect to a variable growth exponent p(x). The proof is based on a modification of the Caccioppoli inequality, which enables us to use existing versions of the Moser iteration. idivA(x;u;ru) = B(x;u;ru); where A and B satisfy natural simple structural conditions with respect to a variable growth exponent p(x); see Theorem 3.5 below. The novelty in our argumentation lies in the choice of test functions. We are able to prove under modified assumptions on the test functions exactly the same Caccioppoli estimate as in the case of p(x)- Laplacian idiv i p(x)jru(x)j p(x)i2 ru(x) ¢ = 0: The point is that the Moser iteration technique used in (8) remains valid under our consideration.