| 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. |
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