We consider the sub- or supercritical Neumann elliptic problem $-\Delta u+\mu u=u^{5+\epsilon}$, $u>0$ in $\Omega $; $\frac{\partial u}{\partial n}=0$ on $\partial\Omega$, $\Omega$ being a smooth bounded domain in $\mathbb{R}^{3}$, $\mu>0$ and $\epsilon\neq0$ a small number. $H_{\mu}$ denoting the regular part of the Green's function of the operator $-\Delta +\mu$ in $\Omega$ with Neumann boundary conditions, and $\varphi_{\mu}(x)=\mu^{\frac{1}{2}}+H_{\mu}(x,x)$, we show that a nontrivial relative homology between the level sets $\varphi_{\mu}^{c}$ and $\varphi_{\mu}^{b}$, $b_0$ small enough, of a solution to the problem, which blows up as $\epsilon$ goes to zero at a point $a\in\Omega$ such that $b\leq\varphi_{\mu}(a)\leq c$. The same result holds, for $\epsilon< 0$, assuming that $0