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){−Δu+a(x)⋅∇u+V(x)u=up+γuq in Ωu=0 on ∂Ω, and(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 RN in the case of N≥4, p>N+1N−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+2N−2<p<N+1N−3 under further symmetry assumptions on Ω, a and V.