We study the existence of multiple solutions for the problem \begin{alignat*}{2} -\Delta u+\lambda u & =|u|^{2^\ast-2}u+f \quad& \text{in }\varOmega, \\ \dfrac{\partial u}{\partial\nu} & =0 & & \text{on }\partial\varOmega, \end{alignat*} and we show that at least one of them is sign changing. Here $\varOmega$ is a bounded domain in $\mathbb{R}^N$ with $N\geq5$ whose boundary is of class $C^2$, $\partial/\partial\nu$ is the outward normal derivative, and $f\in L^{N/2}(\varOmega)$ whose $L^{2N/(N+2)}(\varOmega)$ norm is sufficiently small.