Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sun, 05 Apr 2020 11:08:17 GMT2020-04-05T11:08:17Z5011Semi-stable and extremal solutions of reaction equations involving the p-laplacianhttp://hdl.handle.net/10316/11373Title: Semi-stable and extremal solutions of reaction equations involving the p-laplacian
Authors: Cabré, Xavier; Sanchón, Manel
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
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/113732006-01-01T00:00:00Z