We consider the supercritical elliptic problem $-\Delta u=u^{5+\epsilon},$ $u>0$ in $\Omega $; $u=0$ on $\partial\Omega$ with $\Omega$ a smooth bounded domain in $\mathbb{R}^3$, and $\epsilon>0$ a small number. Denoting by $G$ the Green's function of $-\Delta$ on $\Omega$ with Dirichlet boundary conditions, and by $H$ its regular part, we show that a nontrivial relative homology between the level sets $\varphi^{b}$ and $\varphi^{a}$ of $\varphi$, $0>b>a$, $\varphi(x,y)=H(x,x)^{1/2}H(y,y)^{1/2}-G(x,y)$, implies the existence, for $\epsilon$ small enough, of a solution to the problem which blows up, as $\epsilon$ goes to 0, at two points $x,y$ such that $a\leq\varphi(x,y)\leq b,$ $\nabla\varphi(x,y)=0.$