# On an abstract bifurcation result concerning homogeneous potential operators with applications to PDEs

## TL;DR

This paper investigates bifurcation phenomena for equations involving homogeneous potential operators in Banach spaces, providing theoretical results and applications to various PDEs such as Schrödinger and Kirchhoff equations.

## Contribution

It establishes a bifurcation framework for homogeneous potential operators and applies it to multiple PDE models, describing bifurcation diagrams and parameter estimates.

## Key findings

- Proves bifurcation results using Nehari set analysis.
- Describes full bifurcation diagrams in specific cases.
- Estimates critical parameter values where solutions cease to exist.

## Abstract

We study an abstract equation in a reflexive Banach space, depending on a real parameter $\lambda$. The equation is composed by homogeneous potential operators. By analyzing the Nehari sets, we prove a bifurcation result. In some particular cases we describe the full bifurcation diagram, and in general, we estimate the parameter $\lambda_b$ for which the problem does not have non-zero solution when $\lambda>\lambda_b$. We give many applications to partial differential equations: Kirchhoff type equations, Schr\"odinger equations coupled with the electromagnetic field, Chern-Simons-Schr\"odinger systems and a nonlinear eigenvalue problem.