A complete scenario on nodal radial solutions to the Brezis Nirenberg problem in low dimensions

Abstract

In this paper we consider nodal radial solutions of the problem $$\{\begin{array}{ll}-\Delta u=|u{|}^{2^{\ast }-2}u+\lambda u& \text{ in }B,\\ u=0& \text{ on }\partial B\end{array}$$ where $2^{\ast }=\frac{2N}{N-2}$ with $3\le N\le 6$ and $B$ is the unit ball of ${\mathbb{R}}^N$. We compute the asymptotics of the solution $u$ as well as $||u|{|}_{\infty }$, its first zero and other relevant quantities as $ł$ goes to a critical value $\overline{ł}$. Also the sign of $ł-\overline{ł}$ is established in all cases. This completes an analogous analysis for $N\ge 7$ given in [EGPV].