# Non-uniqueness of blowing-up solutions to the Gelfand problem

## TL;DR

This paper demonstrates the existence of multiple blowing-up solutions to the Gelfand problem on planar domains, using advanced mathematical techniques to analyze solution multiplicity under specific conditions.

## Contribution

It provides the first examples of solution multiplicity for the Gelfand problem with blow-up behavior at a fixed point, employing a refined Lyapunov-Schmidt reduction.

## Key findings

- Multiple solutions with blow-up at the same point are possible.
- The degree of a finite-dimensional map is computed to establish multiplicity.
- Conditions on the potential influence solution behavior.

## Abstract

We consider the Gelfand problem on a planar domain. Under some conditions on the potential, we provide the first examples of multiplicity for blowing-up solutions at a given point in the domain. The argument is based on a refined Lyapunov-Schmidt reduction and the computation of the degree of a finite-dimensional map.