Entropy solutions for the p(x)-Laplace equation

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 exponentCMUC/FCT and MCYT grants BMF2002- 04613-C03, MTM2005-07660-C02 (first author); CMUC/FCT and Project POCI/MAT/57546/2004 (second author

Regularity of entropy solutions of quasilinear elliptic problems related with Hardy-Sobolev inequalities

CMUC/FCT; MEC Spanish grant MTM2004-02223, MTM2004-02223, BFM2003-03772, BMF2002-04613- C03, MTM2005-07660-C02-0

The obstacle problem for nonlinear elliptic equations with variable growth and L1-data

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 L

Boundedness of the extremal solution of some p-Laplacian problems

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.http://www.sciencedirect.com/science/article/B6V0Y-4KFV38H-1/1/cea49519619442ca6f3831d7928ae4e

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 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 < 2MCYT, MEC Spanish grants BMF2002-04613-C03, MTM2005-07660-C02-01; CMUC/FC

Entropy solutions for the p(x)-Laplace equation

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 exponentCMUC/FCT and MCYT grants BMF2002- 04613-C03, MTM2005-07660-C02 (first author); CMUC/FCT and Project POCI/MAT/57546/2004 (second author

Entropy solutions for the p(x)p(x)-Laplace equations

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 $L^1$ data, 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, for which we obtain new inclusion results of independent interest

Entropy solutions for the p(x)p(x)-Laplace equations

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 $L^1$ data, 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, for which we obtain new inclusion results of independent interest