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 R-N with zero Dirichlet boundary conditions. We introduce the notion of semistability 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(cs). Here, m(cs) = m(cs) (N, p) is an explicit exponent which is optimal for the boundedness of semistable solutions. In particular, it is bigger than the critical Sobolev exponent p*-1. We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal L-infinity estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and 1 < p < 2.