The bifurcation diagram of an elliptic Kirchhoff-type equation with respect to the stiffness of the material

TL;DR

This paper analyzes how the solutions of a nonlinear Kirchhoff-type equation change with the material's stiffness parameter, revealing bifurcation phenomena, solution multiplicity, and asymptotic behaviors.

Contribution

It characterizes critical parameters for solution existence and multiplicity in a Kirchhoff equation depending on a real parameter, linking solutions to Sobolev constants.

Findings

Existence of multiple solutions including a local minimum and mountain pass solution.

Identification of a threshold parameter where the local minimum's energy becomes non-negative.

Asymptotic analysis of solutions as the parameter approaches zero.

Abstract

We study a superlinear and subcritical Kirchhoff type equation which is variational and depends upon a real parameter $\lambda$. The nonlocal term forces some of the fiber maps associated with the energy functional to have two critical points. This suggest multiplicity of solutions and indeed we show the existence of a local minimum and a mountain pass type solution. We characterize the first parameter $\lambda_0^*$ for which the local minimum has non-negative energy when $\lambda\ge \lambda_0^*$. Moreover we characterize the extremal parameter $\lambda^*$ for which if $\lambda>\lambda^*$, then the only solution to the Kirchhoff equation is the zero function. In fact, $\lambda^*$ can be characterized in terms of the best constant of Sobolev embeddings. We also study the asymptotic behavior of the solutions when $\lambda\downarrow 0$.