Multiple Sign-Changing Solutions for Kirchhoff-Type Equations - Xingping Li; Xiumei He (Discrete Dynamics in Nature and Society)
We study the following Kirchhoff-type equations -a+b∫Ω∇u2dxΔu+Vxu=fx,u, in Ω, u=0, in ∂Ω, where Ω is a bounded smooth domain of RN  (N=1,2,3), a>0, b≥0, f∈C(Ω¯×R,R), and V∈C(Ω¯,R). Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, if f is odd with respect to its second variable, this problem has infinitely many sign-changing solutions.