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

Abstract

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$-F_i(x,u_i,D{u}_i,D^2{u}_i)-⟨M_i(x)D{u}_i,D{u}_i⟩=\lambda c_{i1}(x)u_1+\cdots +\lambda c_{in}(x)u_n+h_i(x),$$ for $i=1,\cdots ,n$, in a bounded $C^{1,1}$ domain $\Omega \subset {\mathbb{R}}^N$ with Dirichlet boundary conditions; here $n\ge 1$, $\lambda \in \mathbb{R}$, $c_{ij},h_i\in L^{\infty }(\Omega )$, $c_{ij}\ge 0$, $M_i$ satisfies $0<{\mu }_{1}I\le M_i\le {\mu }_{2}I$, 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.

24 pages