Multiplicity solutions and qualitative properties for some classes of nonlinear elliptic equations and Sobolev embeddings on limit case

• Grant number: 23/07697-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: June 01, 2024
• End date: May 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Stefano Nardulli
• Grantee: Raoní Cabral Ponciano
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil
• Associated scholarship(s): 25/07027-9 - Isoperimetric Clusters on Riemannian and Finsler Manifolds: Properties and Applications via the Photographic Method, BE.EP.PD

Abstract

In this project, we would like to research the Van der Waals-Allen-Cahn-Hilliard equation and obtain multiple solutions for the systems involving this equation with a volume constraint using a photography map. One of our goals is to develop a Sobolev inequality with weights for non-radial functions using a weighted isoperimetric inequality. Also, we would appreciate developing some inequalities and properties for the weighted Sobolev space $X^{k,p}_\infty$. Precisely, supercritical Adams' inequalities on weighted Sobolev spaces $X^{k,p}_R$ with $R<\infty$ and $R=\infty$. In addition, we study some qualitative properties for solutions (subcritical and critical) of second order problems on weighted Sobolev spaces for operators in great generality. Our last research within this project involves studying an embedding that incorporates radial functions from a Sobolev space in the hyperbolic space.