Properties of groundstates of nonlinear Schrödinger equations under a weak constant magnetic field
arXiv:1607.00170 · doi:10.1016/j.matpur.2018.05.007
Abstract
We study the qualitative properties of groundstates of the time-independent magnetic semilinear Schrödinger equation \[ - (\nabla + i A)^2 u + u = |u|^{p-2} u, \qquad \text{ in } \mathbb{R}^N, \] where the magnetic potential $A$ induces a constant magnetic field. When the latter magnetic field is small enough, we show that the groundstate solution is unique up to magnetic translations and rotations in the complex phase space, that groundstate solutions share the rotational invariance of the magnetic field and that the presence of a magnetic field induces a Gaussian decay. In this small magnetic field régime, the corresponding ground-energy is a convex differentiable function of the magnetic field.
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