Instability theory of stationary solutions for nonlinear Schrödinger equations

We study the orbital instability of algebraically decaying stationary solutions of nonlinear Schrödinger equations in dimensions d≥2, taking competing double-power nonlinearities as a canonical model. At zero frequency, the essential spectrum of the linearized operator touches zero, and the usual H¹ coercivity is lost. We construct the zero-frequency limit η₀ of the frequency derivatives ∂_ωϕ_ω of standing-wave profiles and determine its sharp asymptotic behavior. For a stationary ground state ϕ₀∈ H¹(Rᵈ), we prove instability if x\,ϕ₀∉ L²(Rᵈ) or, when x\,ϕ₀∈ L²(Rᵈ), if the duality pairing η₀,ϕ₀ is negative. This criterion is a zero-frequency counterpart of the negative slope condition in the Grillakis--Shatah--Strauss theory. The proof combines variation of parameters for the linearized ODE with a shooting analysis that determines the global behavior of the homogeneous solution regular at the origin. We then localize η₀ to construct an unstable direction and establish instability through a Lyapunov functional argument based on the variational characterization of ground states. The results extend with essentially the same proofs to a class of multiple-power nonlinearities.