Nonlocal elliptic Kirchhoff-type problems with concave-convex nonlinearities and changing sign

• Grant number: 25/12051-6
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: October 01, 2025
• End date: September 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Olimpio Hiroshi Miyagaki
• Grantee: Eduardo Dias Lima
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 22/16407-1 - TESdE: Thematic on Equations and Systems of differential Equations, AP.TEM

Abstract

In this project, we will study a class of elliptic problems in Partial Differential Equations (PDEs). The study will be divided into two distinct parts. In the first, we will investigate the existence and multiplicity of positive solutions to an elliptic Kirchhoff-type problem with critical growth and a concave-convex nonlinearity in RN . To find the solutions to this problem, we will use the techniques and methods developed in [26, 53], together with the Nehari method [46, 47] and the nonlinear Rayleigh quotient method [26, 29]. In the second part, we will study the existence of nontrivial solutions to quasilinear and semilinear Schrödinger equations involving potentials and weights with sign change, so that the spectrum of the operator involved presents a negative part in RN . The main approach to find these solutions will be to employ the new method developed in [38] and apply spectral theory together with the classical Linking Theorem, under the Cerami condition, choosing appropriately the function that defines the Linking geometry. It is important to highlight that each of the mentioned problems will be approached with different techniques, which makes our project particularly interesting and challenging.