Generalized Harnack Inequality for Nonhomogeneous Elliptic Equations

This paper is concerned with nonlinear elliptic equations in nondivergence form F(D(2)u, Du, x) = 0 where F has a drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical Harnack inequality. This paper presents a new generalization of the Harnack inequality for such equations. As a corollary we obtain the optimal Harnack type of inequality for p(x)-harmonic functions which quantifies the strong minimum principle.