Spectral estimates of least energy solutions to the Brezis–Nirenberg problem with a variable coefficient

Let uεuε be a least energy solution to the Brezis–Nirenberg problem:−Δu=c0up+εk(x)uin Ω,u>0in Ω,u|∂Ω=0, where Ω⊂RNΩ⊂RN(N⩾6)(N⩾6) is a smooth bounded domain, k∈C2(Ω¯) is a nonnegative function, c0=N(N−2)c0=N(N−2), p=(N+2)/(N−2)p=(N+2)/(N−2) is the critical Sobolev exponent and ε>0ε>0 is a small parameter.We prove several asymptotic estimates of eigenvalues λi,ελi,ε and corresponding eigenfunctions vi,εvi,ε to the eigenvalue problem:{−Δvi,ε=λi,ε(c0puεp−1+εk(x))vi,εin Ω,vi,ε=0on ∂Ω,‖vi,ε‖L∞(Ω)=1 as ε→0ε→0, for i=1,2,…,N+1,N+2i=1,2,…,N+1,N+2.