Nonlinear Elliptic Equations with Singular Potentials

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.