Perturbations of Lane–Emden and Hamilton–Jacobi equations II: Exterior domains

In this article we are interested in the existence of positive classical solutions ofequation(1){−Δu+a(x)⋅∇u+V(x)u=up+γuq in Ωu=0 on ∂Ω, andequation(2){−Δu+a(x)⋅∇u+V(x)u=up+γ|∇u|q in Ωu=0 on ∂Ω, where Ω is a smooth exterior domain in RNRN in the case of N≥4N≥4, p>N+1N−3 and γ∈Rγ∈R. We assume that V   is a smooth nonnegative potential and a(x)a(x) is a smooth vector field, both of which satisfy natural decay assumptions. Under suitable assumptions on q we prove the existence of an infinite number of positive classical solutions.We also consider the case of N+2N−2