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

The V\'azquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients