(From author’s abstract.) Let \(\Omega\subset\mathbb{R}^ N\) be a bounded open set with smooth boundary and \(p=2N/(N-2)\) be the critical Sobolev exponent. We prove the existence of nodal solutions (i.e. solutions which change sign) for the Dirichlet problem \(-\Delta u=| u|^{p- 2}u+\lambda u\) on \(\Omega\) and \(u=0\) on \(\partial\Omega\), when \(N\geq 6\) and \(\lambda\in(0,\lambda_ 1)\), with \(\lambda_ 1\) the first eigenvalue of \(-\Delta\) in \(H^ 1_ 0(\Omega)\).