Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves

For N≥3 and $p>1$, we consider the nonlinear Schrödingerequation i∂ₜw+Δₓw+V(x) |w|ᵖ⁻¹w=0 wherew=w(t,x):RN with a potential V that decays at infinity like |x|⁻ᵇ for some b∈ (0,2). A standing wave is asolution of the form w(t,x)=eⁱu(x) where λ>0 andu:RN. For $ 1 < p < 1+(4-2b)/(N-2)$, we establish the existence of a C¹-branch ofstanding waves parametrized by frequencies λ in a rightneighbourhood of $0$. We also prove that these standing waves areorbitally stable if $ 1 < p < 1+(4-2b)/N$ and unstable if$1+(4-2b)/N < p < 1+(4-2b)/(N-2)$.