On radial solutions of inhomogeneous nonlinear scalar field equations

We study the existence of radially symmetric solutions u∈H1(ω) of the following nonlinear scalar field equation -δu=g(|x|,u) in ω. Here ω=RN or {x∈RN||x|>R}, N≥2. We generalize the results of Li and Li (1993) [13] and Li (1990) [14] in which they studied the problem in RN and {|x|>R} with the Dirichlet boundary condition. Furthermore, we extend it to the Neumann boundary problem and we also consider the nonlinear Schrödinger equation that is the case g(r,s)=-V(r)s+g~(s).