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

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 assumptio...