Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem

We study the asymptotic behavior, as lambda -> 0, of least energy radial sign-changing solutions u(lambda), of the Brezis-Nirenberg problem {-Delta u = lambda u + vertical bar u vertical bar(2*-2)u in B-1 u = 0 on partial derivative B-1, where lambda > 0, 2* = 2n/n-2 and B-1 is the unit ball of R-n, n >= 7. We prove that both the positive and negative part u(lambda)(+) and u(lambda)(-) concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover, we show that suitable rescalings of u(lambda)(+) and u(lambda)(-) converge to the unique positive regular solution of the critical exponent problem in R-n. Precise estimates of the blow-up rate of vertical bar vertical bar u(lambda)(+/-)vertical bar vertical bar infinity are given, as well as asymptotic relations between vertical bar vertical bar u(lambda)(+/-)vertical bar vertical bar infinity and the nodal radius r(lambda). Finally, we prove that, up to constant, lambda -n-2/2n-8 u(lambda) converges in C-loc(1) (B-1 - {0}) to G(x, 0), where G(x, y) is the Green function of the Laplacian in the unit ball.