Multiple solutions for Kirchhoff type problem near resonance

Abstract

Based on Ekeland's variational principle and the mountain pass theorem, we show the existence of three solutions to the Kirchhoff type problem $$\displaylines{ -\Big(a+b\int_{\Omega}|\nabla u|^2dx \Big) \Delta u =b \mu u^3+f(x,u)+h(x), \quad\text{in } \Omega, \cr u=0, \quad \text{on } \partial \Omega. }$$ Where the parameter $\mu$ is sufficiently close, from the left, to the first nonlinear eigenvalue.

Keywords

• Near resonance
• mountain pass theorem
• Kirchhoff type
• Ekeland's variational principle