Perturbations of Lane-Emden and Hamilton-Jacobi equations: The full space case

In this article we are interested in positive classical solutions of −Δu+a(x)⋅∇u+V(x)u=up+γuq  in  RN, and −Δu+a(x)⋅∇u=up+γ|∇u|q  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 positive classical solutions.