An ergodic control problem arising from the principal eigenfunction of an elliptic operator

Let us consider the following second order quasi-linear partial differential equation: (0.1) -1/2Δ vα+H(x, ∇ vₐ)+α vα=0 with a quadratic growth nonlinear term H(x, ∇ v.) on ∇ vₐ , where α is a positive constant.Such kinds of equations on bounded regions with periodic or Neumann boundary conditions have been studied by several authors (cf.Bensoussan-Frehse [3], Gimbert [6], Lasry [8], and Lions [9] $)$ in connection with ergodic control problems, where the asymptotic behaviour of the solution vₐ of (0.1) as α