Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian

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 (perhaps unbounded). We prove that certain minimizers, or one-sided minimizers, of the energy are semi-stable, and study the 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 $m < m_{c s}$. Here, $m_{c s}=m_{c s}(N,p)$ is an explicit exponent which is optimal for the boundedness of semi-stable solutions. In particular, it is bigger than the critical Sobolev exponent $p^\star-1$. We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal $L^\infty$ estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and $1 < p < 2$.