Sign-changing semi-classical solutions of the Brezís-Nirenberg problems with jump nonlinearities in high dimensions

In this paper, we consider the following elliptic equation:{−ε2Δu+α+u+−α−u−=λ|u|p−2u+|u|2⁎−2uin Ω,u=0on ∂Ω, where Ω⊂RN(N≥4) is a bounded domain, u±=max⁡{±u,0}, α±,λ>0 are constants, ε>0 is a small parameter, and 20 that is sufficiently small by means of the variational method. Furthermore, by combining the elliptic and local energy estimates, we study the concentration behavior of this least-energy sign-changing solution and the location of the positive and negative spikes as ε→0+. Our results partially complete the studies in [6] (2011) in the sense that, due to Sobolev embedding, only subcritical cases are considered in that paper.