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

Abstract

We study the asymptotic behavior, as $$\lambda \rightarrow 0$$λ→0, of least energy radial sign-changing solutions $$u_\lambda $$uλ, of the Brezis–Nirenberg problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u + |u|^{2^* -2}u &{}\quad \hbox {in}\ B_1\\ u=0 &{}\quad \hbox {on}\ \partial B_1, \end{array}\right. \end{aligned}$$-Δu=λu+|u|2∗-2uinB1u=0on∂B1,where $$\lambda >0,\, 2^*=\frac{2n}{n-2}$$λ>0,2∗=2nn-2 and $$B_1$$B1 is the unit ball of $$\mathbb {R}^n,\, n\ge 7$$Rn,n≥7. We prove that both the positive and negative part $$u_\lambda ^+$$uλ+ and $$u_\lambda ^-$$uλ- concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover, we show that suitable rescalings of $$u_\lambda ^+$$uλ+ and $$u_\lambda ^-$$uλ- converge to the unique positive regular solution of the critical exponent problem in $$\mathbb {R}^n$$Rn. Precise estimates of the blow-up rate of $$\Vert u_\lambda ^\pm \Vert _{\infty }$$‖uλ±‖∞ are given, as well as asymptotic relations between $$\Vert u_\lambda ^\pm \Vert _{\infty }$$‖uλ±‖∞ and the nodal radius $$r_\lambda $$rλ. Finally, we prove that, up to constant, $$\lambda ^{-\frac{n-2}{2n-8}} u_\lambda $$λ-n-22n-8uλ converges in $$C_{\mathrm{loc}}^1(B_1-\{0\})$$Cloc1(B1-{0}) to $$G(x,0)$$G(x,0), where $$G(x,y)$$G(x,y) is the Green function of the Laplacian in the unit ball.