On the profile of sign changing solutions of an almost critical problem in the ball

TL;DR

This paper investigates sign-changing solutions to a near-critical elliptic PDE in a ball, revealing two new non-radial solutions with complex bubble structures through Lyapunov-Schmidt reduction.

Contribution

It introduces two novel non-radial solutions with three bubbles, obtained as local extrema of a reduced functional, expanding understanding of solution profiles near criticality.

Findings

Discovered two new non-radial solutions with three bubbles.

Solutions are obtained as local minimum and saddle point, not via global min-max.

Solutions exhibit complex nodal structures and are not radially symmetric.

Abstract

We study the existence and the profile of sign-changing solutions to the slightly subcritical problem $$ -\De u=|u|^{2^*-2-\eps}u \hbox{in} \cB, \quad u=0 \hbox{on}\partial \cB, $$ where $\cB$ is the unit ball in $\rr^N$, $N\geq 3$, $2^*=\frac{2N}{N-2}$ and $\eps>0$ is a small parameter. Using a Lyapunov-Schmidt reduction we discover two new non-radial solutions having 3 bubbles with different nodal structures. An interesting feature is that the solutions are obtained as a local minimum and a local saddle point of a reduced function, hence they do not have a global min-max description.