Existence and multiplicity for elliptic problems with quadratic growth in the gradient

TL;DR

This paper investigates the existence and multiplicity of solutions for elliptic equations with quadratic gradient growth, demonstrating solvability under optimal conditions and showing that solutions are generally not unique.

Contribution

It establishes solvability conditions for a class of elliptic problems with quadratic gradient growth and proves that solutions are typically non-unique, extending previous results on the opposite sign case.

Findings

Elliptic problems with quadratic gradient growth are solvable under optimal hypotheses.

Solutions to these problems are generally not unique.

The case with opposite sign zero order term has been previously well-studied.

Abstract

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.