Existence and orbital stability of standing waves to a nonlinear Schr\"odinger equation with inverse square potential on the half-line

Elek Csobo

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arXiv · 2011.10332

Existence and orbital stability of standing waves to a nonlinear Schr\"odinger equation with inverse square potential on the half-line

Elek Csobo

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Abstract

We investigate the properties of standing waves to a nonlinear Schr\"odinger equation with inverse-square potential on the half-line. We first establish the existence of standing waves. Then, by a variational characterization of the ground states, we establish the orbital stability of standing waves for mass sub-critical nonlinearity. Owing to the non-compactness and the absence of translational invariance of the problem, we apply a profile decomposition argument. We obtain convergent minimizing sequences by comparing the problem to the problem at "infinity" (i.e., the equation without inverse square potential). Finally, we establish orbital instability by a blow-up argument for mass super-critical nonlinearity.

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Elek Csobo. 2020-11-20. Existence and orbital stability of standing waves to a nonlinear Schr\"odinger equation with inverse square potential on the half-line. https://arxiv.org/abs/2011.10332

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