# Sobolev versus H\"older local minimizers in degenerate Kirchhoff type problems

## TL;DR

This paper investigates the geometric properties of functionals in degenerate Kirchhoff problems, revealing how degeneracy affects the existence and multiplicity of solutions through Sobolev and H"older minimizer analysis.

## Contribution

It establishes conditions under which classical minimizer results hold or fail in degenerate Kirchhoff problems, advancing understanding of solution existence.

## Key findings

- Results depend on the degree of degeneracy in the problem.
- Existence of solutions varies with the interaction between nonlocal term and nonlinearity.
- Multiplicity of solutions is demonstrated under certain degeneracy conditions.

## Abstract

In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus H\"older local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is. We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering several different interactions between the nonlocal term and the nonlinearity.