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.

中文翻译：

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证明超线性椭圆PDE先验界的新方法

我们描述了一种基于全局弱Harnack不等式和定量Hopf引理证明正解和超线性椭圆PDE解的先验界的新方法。基于该方法的新结果包括：（i）在整个边界上没有边界条件的方程；（ii）具有无限远处没有精确增长的非线性方程；（iii）符号相反的不等式系统。