Least energy sign-changing solutions of the singularly perturbed Brezis–Nirenberg problem

Abstract

Consider the following elliptic equation: −ε2Δu+u=λ|u|p−2u+|u|2∗−2uin Ω,u=0on ∂Ω,where Ω⊂RN(N≥3) is a bounded domain, λ>0 is a constant, ε>0 is a small parameter and 2<p<2∗=2NN−2. By the variational method we study the existence of the least energy sign-changing solution to this problem for ε>0 small enough. Moreover, we investigate the concentration behavior of these solutions and, by combining the elliptic estimates and local energy estimates, we also obtain the location of the positive and negative spikes as ε→0+. Our results partially complete the studies in Bartsch and Weth (2005) and Noussair and Wei (1997) in the sense that due to the Sobolev embedding, only subcritical cases are considered in these two papers.