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

In this paper, we study the existence of sign-changing (nodal) and nontrivial solutions for the nonlinear Kirchhoff-type equation {-(a+b integral(Omega)vertical bar del u vertical bar(2)dx) Delta u = alpha u + beta u(3) in Omega, u=0 on partial derivative Omega, where alpha, beta is an element of R are two real parameters. With the help of nodal Nehari set, we first provide a description of a two-dimensional set in the (alpha,beta) 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. (C) 2019 Elsevier Inc. All rights reserved.