Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries

Abstract

We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem $$-\Delta u=|u|^{2^*-2-\e}u \hbox{in}\Omega, \quad u=0 \hbox{on}\partial \Omega,$$ where $\Omega$ is a smooth bounded domain in $\R^N$, $N\geq 3$, $2^*=\frac{2N}{N-2}$ and $\e>0$ is a small parameter. In particular we prove that if $\Omega$ is convex and satisfies a certain symmetry, then a nodal four-bubble solution exists with two positive and two negative bubbles.