# A new method of proving a priori bounds for superlinear elliptic PDE

## TL;DR

This paper introduces a novel approach for establishing a priori bounds in superlinear elliptic PDEs using global weak Harnack inequalities and a quantitative Hopf lemma, applicable to complex boundary and nonlinearity conditions.

## Contribution

The paper presents a new method for proving a priori bounds that handles boundary-less equations, nonlinearities with unspecified growth, and systems with opposing signs.

## Key findings

- Applicable to equations without boundary conditions
- Handles nonlinearities with non-standard growth
- Effective for systems of inequalities with opposite signs

## Abstract

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.