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

We study the asymptotic behavior, as $lambda ightarrow 0$, of least energy radial sign-changing solutions $u_lambda$, of the Brezis--Nirenberg problem $$-Delta u = lambda u + |u|^{2^* -2}u quad hbox{in} B_1$$ $$u=0 quad hbox{on} partial B_1,$$ where $lambda >0$, $2^*=rac{2n}{n-2}$ and $B_1$ is the unit ball of $R^n$, $ngeq 7$. 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 $|u_lambda^pm|_{infty}$ are given, as well as asymptotic relations between $|u_lambda^pm|_{infty}$ and the nodal radius $r_lambda$. Finally, we prove that, up to constant, $lambda^{-rac{n-2}{2n-8}} u_lambda$ 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.