Speaker : Dr. Rakesh Arora

Title of the Talk : A large class of nonlocal elliptic equations with singular nonlinearities

Date and Time : 25th October 2023, 5:00PM

About the Speaker : Assistant Professor, Department of Mathematics, IIT-BHU

Abstract : In this talk, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of Lsγu = F(u), posed in a smooth bounded domain Ω ⊂ RN with appro-priate homogeneous Dirichlet or outer boundary conditions. The operator Lsγ belongs to a general class of nonlocal operators, including typical fractional Laplacians, restricted fractional Laplacian (RFL), censored fractional Laplacian (CFL), and spectral fractional Laplacian (SFL). The nonlinear term F(u) covers three different amalgamations of singular nonlinearities with singular exponent q > 0, in particular, F(u) ∼ u−q purely singular nonlinearity, F(u) ∼ u −q +f(u) singular nonlinearity with a source term and F(u) ∼ u−q −g(u) singular nonlinearity with an absorption term. Based on a precise analysis of the Green kernel, we developed a new unifying approach that empowered us to construct a theory for nonlocal elliptic equations involving singular nonlinearities. In particular, we show the existence of critical singular exponents q ∗ s,γ and q ∗∗ s,γ, which provides a fairly complete classification of nonlocal elliptic equations with singular nonlinearities via subtle boundary behavior of the weak-dual solution. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.

This is joint work with

1. Prof. Vicent ̧iu D. R Ì?adulescu, Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Krakow, Poland.

Department of Mathematics, University of Craiova, 200585 Craiova, Romania.

Simion Stoilow Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania

2. Prof. Phouc-Tai Nguyen, Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic