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

AbstractThe 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), where V decays at infinity like |x|−b for some b∈(0,2) and r is a perturbation having the same qualitative behaviour as V(x)|s|p−1s for small |s|. f is possibly singular at the origin 0∈RN. A standing wave is a solution of the form w(t,x)=eiλtu(x) where λ>0 and u:RN→R. For 1<p<1+(4−2b)/(N−2), the existence in H1(RN) of a C1-branch of standing waves parametrized by frequencies λ in a right neighbourhood of λ=0 is proven. These standing waves are shown to be orbitally stable if 1<p<1+(4−2b)/N and unstable if 1+(4−2b)/N<p