We consider nonnegative solutions of $-\Delta_p u=f(x,u)$, where $p>1$ and $\Delta_p$ is the $p$-Laplace operator, in a smooth bounded domain of $\mathbb{R}^N$ with zero Dirichlet boundary conditions. We introduce the notion of semi-stability for a solution $u$, and we give examples and properties of this class of solutions. Under some assumptions on $f$ that make its growth comparable to $u^m$, we prove that every semi-stable solution is bounded if $m

We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal $L^\infty$ estimates.