# Asymptotic behavior and existence of solutions for singular elliptic equations

## TL;DR

This paper investigates the asymptotic behavior and existence of solutions for singular semilinear elliptic equations as the parameter b3 tends to infinity, providing new insights into solution limits and optimal existence conditions.

## Contribution

It introduces new results on the asymptotic limits of solutions and establishes optimal existence criteria for related singular elliptic equations under different assumptions on the data.

## Key findings

- Identifies the limit behavior of solutions as b3    infinity.
- Provides conditions for the existence of solutions based on the positivity of the function f.
- Derives optimal existence results for a related elliptic equation involving gradient terms.

## Abstract

We study the asymptotic behavior, as $\gamma$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$ -\Delta u=\frac{f(x)}{u^\gamma}\,\text{ in }\Omega, $$ where $\Omega$ is an open, bounded subset of $\RN$ and $f$ is a bounded function. We deal with the existence of a limit equation under two different assumptions on $f$: either strictly positive on every compactly contained subset of $\Omega$ or only nonnegative. Through this study we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated to $$ -\Delta v + \frac{|\nabla v|^2}{v} = f\,\text{ in }\Omega. $$