  We develop a new, unified approach to the following two classical questions
on elliptic PDE: the strong maximum principle for equations with non-Lipschitz
nonlinearities, and the at most exponential decay of solutions in the whole
space or exterior domains. Our results apply to divergence and nondivergence
operators with locally unbounded lower-order coefficients, in a number of
situations where all previous results required bounded ingredients. Our
approach, which allows for relatively simple and short proofs, is based on a
(weak) Harnack inequality with optimal dependence of the constants in the
lower-order terms of the equation and the size of the domain, which we
establish.
