Existence results of positive solutions for Kirchhoff type equations via bifurcation methods

TL;DR

This paper proves the existence of positive solutions for a class of Kirchhoff type equations using bifurcation methods, under specific conditions on parameters and nonlinearities in bounded domains.

Contribution

It introduces a novel application of bifurcation techniques to establish positive solutions for Kirchhoff equations with nonlinear terms.

Findings

Existence of positive solutions under certain parameter conditions

Application of bifurcation methods to nonlinear PDEs

Conditions on nonlinearity g and parameters for solutions

Abstract

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array} \right. \end{equation*} in a bounded and smooth domain $\Omega$ in ${\rm I}\hskip -0.85mm{\rm R}$. By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters $a,b,p,r$ and the nonlinearity $g$, the existence of positive solutions.