First eigenvalue and maximum principle for fully nonlinear singular operators

In this paper we introduce the notion of first eigenvalue for fully nonlinear operators which are nonvariational but homogeneous. It is unnecessary to emphasize the importance of knowing the spectrum of a linear operator. When the operator is a uniformly elliptic operator of second order Lu = tr(A(x)D2u) associated with a Dirichlet problem in a bounded domain Ω the spectrum is a point spectrum bounded from below and the first eigenvalue λ is paramount. It is well known that λ is positive and it satisfies: • There exists a positive function φ satisfying { Lφ + λφ = 0 in Ω φ = 0 on ∂Ω.