Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave–convex right-hand side

We deal with existence, non-existence and multiplicity of solutions to the model problemequation(P){F(∇uλ,D2uλ)=λuλq+uλr,inΩ,uλ>0,inΩ,uλ=0,on∂Ω, where Ω⊂RnΩ⊂Rn is a smooth bounded domain, F   is a 1-homogeneous fully nonlinear uniformly elliptic operator, λ>0λ>0, 0