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

We consider the problem of finding lambda is an element of R and a function u : R-n -> R that satisfy the PDE max {lambda + F (D(2)u) - f(x), H (Du)} = 0, x is an element of R-n. 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 vertical bar x vertical bar -> infinity. 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."