A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

TL;DR

This paper investigates fully nonlinear elliptic equations with quadratic gradient growth, establishing a priori bounds and multiplicity results, extending previous divergence-form findings to nondivergence form using maximum principle techniques.

Contribution

It introduces a new method for uniform a priori bounds for fully nonlinear equations with quadratic growth, enabling degree theory application and extending multiplicity results beyond divergence form.

Findings

Established uniform a priori bounds for solutions.

Proved multiplicity of solutions in noncoercive cases.

Extended phenomena observed in divergence form to fully nonlinear equations.

Abstract

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as $$ -F(x,u,Du,D^2u) =\lambda c(x)u+\langle M(x)D u, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundary condition, here $\lambda \in\mathbb{R}$, $c,\, h \in L^p(\Omega)$, $p>n\geq 1$, $c\gneqq 0$ and the matrix $M$ satisfies $0<\mu_1 I\leq M\leq \mu_2 I$. Recently this problem was studied in the "coercive" case $\lambda c\le0$, where uniqueness of solutions can be expected, and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle. We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a V\'azquez type strong maximum principle for our kind of equations.