On radial solutions of inhomogeneous nonlinear scalar field equations

We study the existence of radially symmetric solutions u ∈ H 1 ( Ω ) of the following nonlinear scalar field equation − Δ u = g ( | x | , u ) in Ω. Here Ω = R N or { x ∈ R N | | 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 R N 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 ) .