Symmetry Breaking for Ground States of Biharmonic Nonlinear Schrödinger Equations
We consider ground states solutions, suitably defined as energy minimizers, of a class of semilinear biharmonic (fourth-order) Schrödinger equations. By exploiting a connection to the adjoint Stein--Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a symmetry breaking result for ground state solutions, which is in striking contrast to the well-known results of radial symmetry for ground states of classical second-order nonlinear Schrödinger equations. We also discuss symmetry breaking for a minimization problem with constrained mass and for a related problem on the unit ball subject to Dirichlet boundary conditions.
This is joint work with Enno Lenzmann (Universität Basel).