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

Abstract

In this paper we address the following Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} -\varDelta (g(|\nabla u|_2^2) u + u^r) = a u + b u^p&{} \quad \text{ in }~\varOmega , u>0&{}\quad \text{ in }~\varOmega , u= 0&{} \quad \text{ on }~\partial \varOmega , \end{array} \right. \end{aligned}$$ in a bounded and smooth domain $$\varOmega $$ in $$\mathbb {R}^{N}$$ . By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters a, b, p, r and the non-linearity g, the existence of positive solutions.