Solutions to an inhomogeneous equation involving Aronsson operator

Let F : R n → [ 0 , + ∞ ) be a convex function of class C 2 ( R n ∖ { 0 } ) , which is positively homogeneous of degree 1, we assume further that F ( p ) > 0 for any p ≠ 0 , and Hess ( F 2 ) is positive definite in R n ∖ { 0 } . In this paper, for a bounded domain Ω ⊂ R n , f ∈ C ( Ω ) with inf Ω f ( x ) > 0 and g ∈ C ( ∂ Ω ) , we obtain existence and uniqueness results of viscosity solutions to the Dirichlet boundary value problem for a nonlinearly highly degenerate elliptic equation of the form { 1 [ F ( D u ) ] h A u = f , in  Ω u = g , on  ∂ Ω where A denotes the Aronsson operator given by A u = ∑ i , j = 1 n ∂ 2 u ∂ x i ∂ x j ∂ ( 1 2 F 2 ) ∂ p i ( D u ) ∂ ( 1 2 F 2 ) ∂ p j ( D u ) and 0 ≤ h < 2 .