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

Let uε be a least energy solution to the Brezis–Nirenberg problem: where (N 6) is a smooth bounded domain, is a nonnegative function, c0=N(N−2), p=(N+2)/(N−2) is the critical Sobolev exponent and ε>0 is a small parameter. We prove several asymptotic estimates of eigenvalues λi,ε and corresponding eigenfunctions vi,ε to the eigenvalue problem: as ε→0, for i=1,2,…,N+1,N+2.