Large solutions of semilinear equations with Hardy potential

TL;DR

This paper investigates the existence and uniqueness of large solutions to semilinear equations involving Hardy potentials, specifically for equations with nonlinearities satisfying Keller-Osserman conditions in smooth domains.

Contribution

It establishes conditions under which large solutions exist and are unique for equations with Hardy potentials and nonlinearities like power and exponential types.

Findings

Existence of large solutions under certain conditions.

Uniqueness of solutions when parameters satisfy specific criteria.

Applicability to power and exponential nonlinearities.

Abstract

We consider equations of the form $-L_\mu u +f(u)=0$ in a smooth domain $\Omega$, where $L_\mu=\Delta + \mu\delta^{-2}$ and $\delta(x)$ denotes the distance of the point $x$ to the boundary of the domain. The nonlinear term $f$ is positive, increasing and convex on $(0,\infty)$, satisfies the Keller-Osserman condition and some additional technical assumptions. The conditions are satisfied, in particular, by power and exponential nonlinearities. We discuss the question of existence and uniqueness of large solutions when $\mu>0$.