Nonlinear Elliptic Equations with Singular Potentials

• Grant number: 12/10153-6
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: November 01, 2012
• End date: July 31, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Lucas Catão de Freitas Ferreira
• Grantee: Sérgio Leandro Nascimento Neves
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

In this project we study two problems about elliptic partial differential equations with critical singular potential of Hardy type. They can be either isotropic or anisotropic, and polar (one singularity) or multipolar (finitely many singularities). In the first problem, we consider the potential and nonlinearity in the interior of a smooth domain, handling it by perturbation methods and Lyapunov-Schmidt reductions. In the second one, the potential, singularities and nonlinearity are on the boundary of the domain, and we intend to study it by means of the following approaches: minimization methods combined with Hardy inequality on the boundary; or fixed point arguments with integral formulations based on Green functions. We investigate the existence and uniqueness of solutions and their asymptotic behaviors near the singularities of the potential. Another considered approach for the two problems is through optimal mass transport techniques, in order to find solutions as steady states and asymptotic limits of the corresponding parabolic models.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

GLADIALI, FRANCESCA; GROSSI, MASSIMO; NEVES, SERGIO L. N.. Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, v. 18, n. 5, OCT 2016. (12/10153-6)

GLADIALI, FRANCESCA; GROSSI, MASSIMO; NEVES, SERGIO L. N.. Nonradial solutions for the Henon equation in R-N. ADVANCES IN MATHEMATICS, v. 249, p. 1-36, DEC 20 2013. (12/10153-6)

FERREIRA, LUCAS C. F.; NEVES, SERGIO L. N.. ON ELLIPTIC EQUATIONS WITH SINGULAR POTENTIALS AND NONLINEAR BOUNDARY CONDITIONS. QUARTERLY OF APPLIED MATHEMATICS, v. 76, n. 4, p. 699-711, DEC 2018. (12/10153-6)