Let $B$ be the unit ball of $\mathbb{R}^n$, $n \ge 3$. We consider the problem $\Delta u = f(\vert x\vert)u^{p-\epsilon}$ in $B$, $u > 0$ in $B$, $u = 0$ on $\partial B$, where $f \in C^\infty(\mathbb{R},\mathbb{R})$, $p = (n+2)/(n-2)$, $\epsilon \ge 0$. First, we study the behavior of the minimizing radially symmetric solutions of the problem when $\epsilon \to 0^+$. According to the (local or global) monotony of $f$, they converge to a solution of the critical problem or they blow up. The two cases are described. As a consequence, for large $n$ and all $\epsilon > 0$, the critical problems with $f(r)=1+\epsilon r^k$ have minimizing radially symmetric solutions. They necessarily blow up when $\epsilon \to 0^+$. Here again, we describe the blow up.