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

We study the asymptotic behavior, as \(\lambda \rightarrow 0\), of least energy radial sign-changing solutions \(u_\lambda \), of the Brezis–Nirenberg problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u + |u|^{2^* -2}u &{}\quad \hbox {in}\ B_1\\ u=0 &{}\quad \hbox {on}\ \partial B_1, \end{array}\right. \end{aligned}$$ where \(\lambda >0,\, 2^*=\frac{2n}{n-2}\) and \(B_1\) is the unit ball of \(\mathbb {R}^n,\, n\ge 7\). 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 \(\mathbb {R}^n\). Precise estimates of the blow-up rate of \(\Vert u_\lambda ^\pm \Vert _{\infty }\) are given, as well as asymptotic relations between \(\Vert u_\lambda ^\pm \Vert _{\infty }\) and the nodal radius \(r_\lambda \). Finally, we prove that, up to constant, \(\lambda ^{-\frac{n-2}{2n-8}} u_\lambda \) converges in \(C_{\mathrm{loc}}^1(B_1-\{0\})\) to \(G(x,0)\), where \(G(x,y)\) is the Green function of the Laplacian in the unit ball.