Multiplicity of Solutions for an Elliptic Kirchhoff Equation

Abstract In this paper we study the existence of positive solution to Kirchhoff elliptic problem $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\left( 1+\gamma G'\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}\right) \right) \Delta u = \lambda f(u) &{} \text{ in } \; \Omega ,\\ 0 on \partial \end{array}\right. \end{aligned}$$ - 1 + ? G ? ? ? u L 2 ( ? ) ? = ? f in , 0 on ? where $$\Omega $$ is an open, bounded subset $$\mathbb {R}^N$$ R N ( $$N\ge 3$$ ? 3 ), f a locally Lipschitz continuous real function, $$f(0)\ge 0$$ , $$G'\in C(\mathbb {R}^+)$$ ? C and $$G'\ge . We prove at least two solutions with $$L^\infty (\Omega )$$ ? norm between consecutive zeroes for large $$\lambda