  We describe a new method of proving a priori bounds for positive
supersolutions and solutions of superlinear elliptic PDE, based on global weak
Harnack inequalities and a quantitative Hopf lemma. Novel results based on the
method include: (i) equations without a boundary condition on the whole
boundary; (ii) equations with nonlinearities which do not have precise growth
at infinity; (iii) systems of inequalities with opposite sign.
