Symmetric Ground States for Doubly Nonlocal Equations with Mass Constraint

Abstract

We prove the existence of a spherically symmetric solution for a Schrödinger equation with a nonlocal nonlinearity of Choquard type. This term is assumed to be subcritical and satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the Lebesgue space, is prescribed in advance. The approach to this constrained problem relies on a Lagrange formulation and new deformation arguments. In addition, we prove that the obtained solution is also a ground state, which means that it realizes minimal energy among all the possible solutions to the problem.

Keywords

• double nonlocality
• Choquard nonlinearity
• Hartree term
• fractional Laplacian
• nonlinear Schrödinger equation
• normalized solutions