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

Abstract In this article we are interested in the existence of positive classical solutions of (1) { − Δ u + a ( x ) ⋅ ∇ u + V ( x ) u = u p + γ u q  in  Ω u = 0  on  ∂ Ω , and (2) { − Δ u + a ( x ) ⋅ ∇ u + V ( x ) u = u p + γ | ∇ u | q  in  Ω u = 0  on  ∂ Ω , where Ω is a smooth exterior domain in R N in the case of N ≥ 4 , p > N + 1 N − 3 and γ ∈ R . We assume that V is a smooth nonnegative potential and 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 + 2 N − 2 p N + 1 N − 3 under further symmetry assumptions on Ω, a and V .