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

Abstract

We consider nonnegative solutions of −_pu = f(x, u), where p > 1 and _p is the p-Laplace operator, in a smooth bounded domain of RN 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 um, we prove that every semi-stable solution is bounded if m < mcs. Here, mcs = mcs(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_ − 1. We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal L1 estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and 1 < p < 2