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

We study the asymptotic behavior, as $λ\rightarrow 0$, of least energy radial sign-changing solutions $u_λ$, of the Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ B_1\\ u=0 & \hbox{on}\ \partial B_1, \end{cases} \end{equation*} where $λ>0$, $2^*=\frac{2n}{n-2}$ and $B_1$ is the unit ball of $\R^n$, $n\geq 7$. We prove that both the positive and negative part $u_λ^+$ and $u_λ^-$ concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover we show that suitable rescalings of $u_λ^+$ and $u_λ^-$ converge to the unique positive regular solution of the critical exponent problem in $\R^n$. Precise estimates of the blow-up rate of $\|u_λ^\pm\|_{\infty}$ are given, as well as asymptotic relations between $\|u_λ^\pm\|_{\infty}$ and the nodal radius $r_λ$. Finally we prove that, up to constant, $λ^{-\frac{n-2}{2n-8}} u_λ$ 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.