Existence and orbital stability of standing waves for some nonlinear Schrödinger equations, perturbation of a model case

The following nonlinear Schrödinger equation is studiedi∂tw+Δw+f(x,w)=0,w=w(t,x):R×RN→C,N⩾3.f   is a nonlinearity that can be written in the form f(x,s)=V(x)|s|p−1s+r(x,s)f(x,s)=V(x)|s|p−1s+r(x,s), where V   decays at infinity like |x|−b|x|−b for some b∈(0,2)b∈(0,2) and r   is a perturbation having the same qualitative behaviour as V(x)|s|p−1sV(x)|s|p−1s for small |s||s|. f   is possibly singular at the origin 0∈RN0∈RN. A standing wave is a solution of the form w(t,x)=eiλtu(x)w(t,x)=eiλtu(x) where λ>0λ>0 and u:RN→R. For 1