An eigenvalue problem for a fully nonlinear elliptic equation with gradient constraint

Abstract

We consider the problem of finding \(\lambda \in {\mathbb {R}}\) and a function \(u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) that satisfy the PDE $$\begin{aligned} \max \left\{ \lambda + F(D^2u) -f(x),H(Du)\right\} =0, \quad x\in {\mathbb {R}}^n. \end{aligned}$$ Here F is elliptic, positively homogeneous and superadditive, f is convex and superlinear, and H is typically assumed to be convex. Examples of this type of PDE arise in the theory of singular ergodic control. We show that there is a unique \(\lambda ^*\) for which the above equation has a solution u with appropriate growth as \(|x|\rightarrow \infty \). Moreover, associated to \(\lambda ^*\) is a convex solution \(u^*\) that has essentially bounded second derivatives, provided F is uniformly elliptic and H is uniformly convex. It is unknown whether or not \(u^*\) is unique up to an additive constant; however, we verify that this is the case when \(n=1\) or when F, f, H are “rotational.”