Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions

In this paper, we consider the following quasilinear elliptic exterior problem{−div(a(x)|∇u|p−2∇u)+g(x)|u|q−2u=h(x)|u|s−2u+λH(x)|u|r−2u,x∈Ω,a(x)|∇u|p−2∂u∂ν+b(x)|u|p−2u=0,x∈Γ=∂Ω where Ω   is a smooth exterior domain in RNRN, and ν is the unit vector of the outward normal on the boundary Γ  , 1rq>r and at least one solution if 1