Properties of ground states of nonlinear Schrödinger equations under a weak constant magnetic field

Abstract

Abstract We study the qualitative properties of ground states of the time-independent magnetic semilinear Schrodinger equation − ( ∇ + i A ) 2 u + u = | u | p − 2 u in\xa0 R N where the magnetic potential A induces a constant magnetic field. When the latter magnetic field is small enough, we show that the ground state solution is unique up to magnetic translations and rotations in the complex phase space, that ground state 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 regime, the corresponding ground-energy is a convex differentiable function of the magnetic field.