Multiplicity of Solutions for an Elliptic Kirchhoff Equation

Abstract

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