Large solutions to semilinear elliptic equations with Hardy potential and exponential nonlinearity

TL;DR

This paper investigates large solutions to a semilinear elliptic equation with exponential nonlinearity and Hardy potential, establishing bounds and conditions for existence, nonexistence, and uniqueness on bounded domains.

Contribution

It extends previous methods to handle exponential nonlinearities, providing a comprehensive analysis of large solutions with Hardy potentials.

Findings

Derived global a priori bounds of Keller-Osserman type

Established conditions for existence and nonexistence of large solutions

Proved uniqueness of solutions under certain conditions

Abstract

On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the Keller-Osserman type. Using a Phragmen-Lindelof alternative for generalized sub and super-harmonic functions we discuss existence, nonexistence and uniqueness of so-called large solutions, i.e., solutions which tend to infinity at the boundary. The approach develops the one used by the same authors for a problem with a power nonlinearity instead of the exponential nonlinearity.