Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Tue, 20 Aug 2019 14:38:36 GMT2019-08-20T14:38:36Z5051Semi-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/10316/113732006-01-01T00:00:00ZEntropy solutions for the p(x)-Laplace equationhttp://hdl.handle.net/10316/11376Title: Entropy solutions for the p(x)-Laplace equation
Authors: Sanchón, Manel; Urbano, José Miguel
Abstract: We consider a Dirichlet problem in divergence form with variable growth,
modeled on the p(x)-Laplace equation. We obtain existence and uniqueness of an entropy
solution for L1 data, extending the work of B´enilan et al. [5] to nonconstant exponents, as
well as integrability results for the solution and its gradient. The proofs rely crucially on a
priori estimates in Marcinkiewicz spaces with variable exponent
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/113762006-01-01T00:00:00ZRegularity of entropy solutions of quasilinear elliptic problems related with Hardy-Sobolev inequalitieshttp://hdl.handle.net/10316/11340Title: Regularity of entropy solutions of quasilinear elliptic problems related with Hardy-Sobolev inequalities
Authors: Abdellaoui, Boumediene; Colorado, Eduardo; Sanchón, Manel
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/113402006-01-01T00:00:00ZBoundedness of the extremal solution of some p-Laplacian problemshttp://hdl.handle.net/10316/4603Title: Boundedness of the extremal solution of some p-Laplacian problems
Authors: Sanchón, Manel
Abstract: In this article we consider the p-Laplace equation on a smooth bounded domain of with zero Dirichlet boundary conditions. Under adequate assumptions on f we prove that the extremal solution of this problem is in the energy class independently of the domain. We also obtain Lq and W1,q estimates for such a solution. Moreover, we prove its boundedness for some range of dimensions depending on the nonlinearity f.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10316/46032007-01-01T00:00:00ZThe obstacle problem for nonlinear elliptic equations with variable growth and L1-datahttp://hdl.handle.net/10316/11321Title: The obstacle problem for nonlinear elliptic equations with variable growth and L1-data
Authors: Rodrigues, José Francisco; Sanchón, Manel; Urbano, José Miguel
Abstract: The aim of this paper is twofold: to prove, for L1-data, the existence and
uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations
with variable growth, and to show some convergence and stability properties of the corresponding
coincidence set. The latter follow from extending the Lewy–Stampacchia inequalities
to the general framework of L1
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/113212006-01-01T00:00:00Z