# A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems

## TL;DR

This paper investigates the existence, multiplicity, and uniqueness of solutions to a class of elliptic boundary value problems with nonlinear gradient terms and singularities, revealing how singularity strength influences solution behavior.

## Contribution

It introduces a comprehensive analysis of solution multiplicity and uniqueness depending on the severity of the singularity in the nonlinear elliptic problem.

## Key findings

- Existence and multiplicity of solutions for mild singularities.
- Uniqueness of solutions for strong singularities.
- Solution behavior varies with singularity strength.

## Abstract

We study a boundary value elliptic problem having a lower order nonlinear term with subquadratic growth in the gradient of the solution and possibly singular when the solution vanishes. If the singularity is mild enough (and even in the absence of the singularity), we prove an existence and multiplicity result. On the contrary, we prove an existence and uniqueness result for strong singularities.