# On the global bifurcation diagram of the Gel'fand problem

## TL;DR

This paper analyzes the global bifurcation diagram of solutions to the Gel'fand problem in certain domains, revealing new monotonicity properties of non-minimal solutions using energy parametrization and spectral analysis.

## Contribution

It provides the first detailed description of the monotonicity of non-radial, non-symmetric solution branches crossing the origin in the Gel'fand problem.

## Key findings

- First characterization of non-minimal solution branch monotonicity
- Use of energy-based parametrization instead of supremum norm
- Application of spectral analysis to mean field equations

## Abstract

For domains of first kind [7,13] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel'fand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of non-minimal solutions which is not just concerned with radial solutions [28] and/or with symmetric domains [23]. Toward our goal we parametrize the branch not by the $L^{\infty}(\Omega)$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a carefully modified spectral analysis of mean field type equations.