Weighted Trudinger-Moser inequalities on unbounded domains

• Grant number: 17/26582-7
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: May 01, 2018
• End date: April 30, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Djairo Guedes de Figueiredo
• Grantee: José Vitor Pena
• Supervisor: Bernhard Heinrich Ruf
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Università degli Studi di Milano, Italy
• Associated to the scholarship: 16/15887-9 - Aspects of Nonlinear Elliptic Equations and Systems, BP.DR

Abstract

Trudinger-Moser inequality deals with the critical case of the Sobolev embedding: if p=n, \Omega \subset R^n is a C^{1} and bounded domain and u \in W_{0}^{1,p}(\Omega), then for any \alpha > 0, e^{\alpha |u|^q} \in L^{1}(\Omega), for q = \frac{n}{n-1}. It was showed by J. Moser that if \alpha \leq n \omega_{n-1}^{1/n-1}, the supremum (taken over the unit ball of W_{0}^{1,p}(\Omega)) of the integral \int_{\Omega} e^{\alpha |u|^q} is less or equal than a constant times |\Omega| (the measure of \Omega), and if \alpha > n \omega_{n-1}^{1/n-1}, it is infinite. The supremum is attained in any bounded domain: the borderline case (where compacity fails) was first shown by Carleson and Chang for n=2 and \Omega = B_{1}(0).B. Ruf extended the result of Moser for unbounded domains, along with the construction of and explicit sequence of functions (u_j) \in W_{0}^{1,2}(\Omega) such that\int_{\Omega} e^{4 \pi u_j^2} dx converges to the supremum, extending also the result by Carleson and Chang. The main objective of this project is to deal with similar questions for weighted Trudinger-Moser type inequalities.