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

TL;DR

This paper thoroughly analyzes the behavior of nodal radial solutions to a nonlinear elliptic PDE in low dimensions, detailing asymptotic properties and solution characteristics as a parameter approaches a critical value.

Contribution

It provides a complete asymptotic analysis of solutions in dimensions 3 to 6, extending previous work for higher dimensions, and clarifies solution behavior near critical parameters.

Findings

Asymptotic behavior of solutions as λ approaches critical value

Determination of the sign of λ - λ̄ in all cases

Explicit calculations of solution norms and zeros

Abstract

In this paper we consider nodal radial solutions of the problem $$ \begin{cases} -\Delta u=|u|^{2^*-2}u+\lambda u&\text{ in }B,\\ u=0&\text{ on }\partial B \end{cases} $$ where $2^*=\frac{2N}{N-2}$ with $3\le N\le6$ and $B$ is the unit ball of $\R^N$. We compute the asymptotics of the solution $u$ as well as $||u||_\infty$, its first zero and other relevant quantities as $\l$ goes to a critical value $\bar\l$. Also the sign of $\l-\bar\l$ is established in all cases. This completes an analogous analysis for $N\ge7$ given in [EGPV].