Elliptic problems with critical exponential nonlinearities

Abstract

This project addresses the study of a class of elliptic problems. The investigation focuses on finding solutions to partial differential equations (PDEs) of the form Lu(x) + V(x)u = f(x, u), ýx ©, in a domain © ‚ R^n, and some of their properties. Here, Lu(x) is a general elliptic operator, V(x) is a potential, and f(x, u) represents the nonlinearity with critical exponential growth. The study of PDEs is motivated by their application in various scientific fields, from Physics to Economics. The existence of solutions to this type of equation is fundamental to validate the solvability of problems modeled in real-world situations. Additionally, the theory of elliptic operators plays a crucial role in mathematical analysis, being essential for understanding the properties of solutions to elliptic equations. The research aims to explore problems with various elliptic operators, such as the Laplacian, Fractional Laplacian, Integrodifferential operator, and Biharmomic, seeking to contribute to the advancement of the mathematical theory underlying this class of problems. (AU)