A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

TL;DR

This paper establishes uniform a priori bounds and explores the existence and multiplicity of solutions for fully nonlinear elliptic systems with quadratic gradient growth, extending known results to more complex systems with new solution branches.

Contribution

It introduces optimal weak coupling conditions for a priori bounds and demonstrates new solution branches, including in the scalar case, for nonlinear elliptic systems.

Findings

Uniform a priori bounds for the systems.

Existence of solutions under weak coupling.

Discovery of new solution branches in scalar cases.

Abstract

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle =\lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n +h_i(x), $$ for $i=1,\cdots,n$, in a bounded $C^{1,1}$ domain $\Omega\subset \mathbb{R}^N$ with Dirichlet boundary conditions; here $n\geq 1$, $\lambda \in\mathbb{R}$, $c_{ij},\, h_i \in L^\infty(\Omega)$, $c_{ij}\geq 0$, $M_i$ satisfies $0<\mu_1 I\leq M_i\leq \mu_2 I$, and $F_i$ is an uniformly elliptic Isaacs operator. We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.