Sign-changing and nontrivial solutions for a class of Kirchhoff-type problems

Abstract

In this paper, we study the existence of sign-changing (nodal) and nontrivial solutions for the nonlinear Kirchhoff-type equation{−(a+b∫Ω|∇u|2dx)Δu=αu+βu3inΩ,u=0on∂Ω, where α,β∈R are two real parameters. With the help of nodal Nehari set, we first provide a description of a two-dimensional set in the (α,β) plane, which corresponds to the nonexistence and existence of sign-changing solutions for the above Kirchhoff-type equation. And then, we establish the existence result of nontrivial solutions via the minimax methods.