Some classes of Choquard-Schrödinger systems: Existence, multiplicity and properties of solutions

• Grant number: 23/05445-2
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: August 01, 2023
• End date: July 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Ederson Moreira dos Santos
• Grantee: Eduardo de Souza Böer
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

This project is intended to study some classes of Choquard and Choquard-Schröndiger systems in dimensions greater than or equal to 1. We will study the existence and multiplicity of solutions for systems with Riesz kernels and indeterminate kernels with logarithmic growth. We will mainly focus on special solutions, for example of ground state type, minimum energy nodal solutions and solution at Mountain Pass level. Such interest comes from the physical motivations of these solutions, they are the natural solutions for the problems, and also from investigation in the mathematical community for solutions of this type. In addition, we will study properties of these solutions, such as sign, decay and symmetry.The problems that we will address present new difficulties due to the presence of an indeterminate kernel and require the search for new methods, for example regarding limitation and convergence of special sequences of functions linked to the problems. Once such barriers are overcome, these results will contribute to the advance of Choquard systems studies. Furthermore, our assumptions will allow nonlinearities with critical growth, which on the one hand increase the difficulty to prove the existence of solutions, and on the other hand make the results more interesting and general.