Variational problem with repulsive-attractive kernels and its application

In this paper, we continue our previous work [6], [25], [35], and focus on standing waves with prescribed mass for the Hartree equation with Repulsive-attractive kernels, which are used in particle physics to describe the nonlocal interaction among particles [22]. First, we consider a family of interaction functionals consisting of power-law potentials with attractive and repulsive parts and establish the existence of global minimizers. By relaxing the uniform boundedness and radial symmetry conditions, we prove a conjecture raised by Choksi-Fetecau-Topaloglu in [14]. Then as an application, based on classification of attractive part in the kernel, a complete study on existence and qualitative analysis of standing waves for the Hartree equation with repulsive-attractive kernels are given. With respect to the case of single or purely attractive kernels considered in [6], [25], [35], the competition between the two parts in repulsive-attractive kernels forces new implements to catch the solutions and analyze their Lane-Emden (or Hartree) profiles as particles gather (or dissipate).