Instability theory of stationary solutions for nonlinear Schrödinger equations

Abstract

We study the orbital instability of algebraically decaying stationary solutions of nonlinear Schrödinger equations in dimensions $d\ge 2$, taking competing double-power nonlinearities as a canonical model. At zero frequency, the essential spectrum of the linearized operator touches zero, and the usual $H^1$ coercivity is lost. We construct the zero-frequency limit ${\eta }_0$ of the frequency derivatives ${\partial }_{\omega }{\varphi }_{\omega }$ of standing-wave profiles and determine its sharp asymptotic behavior. For a stationary ground state ${\varphi }_0\in H^1({\mathbb{R}}^d)$, we prove instability if $⟨x⟩{\varphi }_0\notin L^2({\mathbb{R}}^d)$ or, when $⟨x⟩{\varphi }_0\in L^2({\mathbb{R}}^d)$, if the duality pairing $⟨{\eta }_0,{\varphi }_0⟩$ 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 ${\eta }_0$ 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.

arXiv comment66 pages, 2 figures