## Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three.(English)Zbl 1322.35122

The article is concerned with stability questions for a 3D Schrödinger equation describing a delta-type point interaction of a strength which depends on the wave function according to a power-type law, implying that the equation is nonlinear. The authors first prove the existence of standing wave solutions with a spatially localized amplitude function (which is computable in closed form). The main goal is then to investigate the stability of these standing waves. The authors prove orbital stability if the exponent in the considered power law is between 0 and 1, and orbital instability if it is larger than or equal to 1. Moreover, they prove asymptotic stability (in an appropriate sense) if the exponent is less than $$1/\sqrt{2}$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35J10 Schrödinger operator, Schrödinger equation 35B35 Stability in context of PDEs
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### References:

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