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

Abstract

We study the asymptotic behavior, as $\lambda \to 0$, of least energy radial sign-changing solutions $u_{\lambda }$, 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 $\lambda >0$, $2^{\ast }=\frac{2n}{n-2}$ 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 $C_{loc}^1(B_1-\{0\})$ to $G(x,0)$, where $G(x,y)$ is the Green function of the Laplacian in the unit ball.

arXiv comment28 pages

References in corpus (2)

• Lane Emden problems with large exponents and singular Liouville equations
• Sign-changing stationary solutions and blowup for the nonlinear heat equation in dimension two