Ground state sign-changing solutions for a class of nonlocal problem

Abstract

In this paper, we study the following nonlocal problem{−(a−λ∫Ω|∇u|2dx)Δu=|u|p−2u,x∈Ω,u=0,x∈∂Ω, where Ω⊂RN (N≥1) is a bounded smooth domain, a>0 is a constant, λ>0 is a parameter and 2<p<2⁎. Combining truncated technique and constraint variational method, we first establish the existence of ground state sign-changing solution uλ for the problem if the parameter λ>0 is sufficiently small. Moreover, we show that the energy of uλ is strictly larger than the ground state energy. Finally, the asymptotic behavior of uλ as λ↘0 is discussed.