Special Session 60: 

Groundstates and radial solutions to Schrodinger-Poisson-Slater equations at the critical frequency

Vitaly Moroz
Swansea University
Wales
Co-Author(s):    Jacopo Bellazzini, Marco Ghimenti, Carlo Mercuri, Jean Van Schaftingen
Abstract:
Schrodinger-Poisson-Slater (SPS) equation is derived as a mean-filed limit of the linear N-body Schrodinger problem for fermionic particles. We discuss the fractional Coulomb-Sobolev function spaces, which appear as the natural domain for the energy functional of a class of equations of SPS type, and establish a family of optimal interpolation inequalities associated with the Coulomb-Sobolev spaces. We also prove the existence of optimizers for the inequalities, which implies the existence of ground-states to SPS equations for a certain range of the parameters. Finally, we discuss radial Strauss type estimates and use them to prove the existence of radial solutions to SPS equations in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities. The latter suggests a striking radial symmetry breakup conjecture.