Differential and Integral Equations

On the behaviour of solutions to the Dirichlet problem for the $p(x)$-Laplacian when $p(x)$ goes to $1$ in a subdomain

Anna Mercaldo, Julio D. Rossi, Sergio Segura de León, and Cristina Trombetti

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper we prove a stability result for some classes of elliptic problems involving variable exponents. More precisely, we consider the Dirichlet problem for an elliptic equation in a domain $\Omega$, which is the $p$--Laplacian equation, $-\mbox{div}(|\nabla u|^{p-2} \nabla u) =f$, in a subdomain $\Omega_1$ of $\Omega$ and the Laplace equation, $-\Delta u = f$, in its complementary (that is, our equation involves the so-called $p(x)$--Laplacian with a discontinuous exponent). We assume that the right-hand side $f$ belongs to $L^\infty(\Omega)$. For this problem, we study the behaviour of the solutions as $p$ goes to $1$, showing that they converge to a function $u$, which is almost everywhere finite when the size of the datum $f$ is small enough. Moreover, we prove that this $u$ is a solution of a limit problem involving the $1$-Laplacian operator in $\Omega_1$. We also discuss uniqueness under a favorable geometry.

Article information

Source
Differential Integral Equations, Volume 25, Number 1/2 (2012), 53-74.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012825

Mathematical Reviews number (MathSciNet)
MR2906546

Zentralblatt MATH identifier
1249.35121

Subjects
Primary: 35J75: Singular elliptic equations 35J66: Nonlinear boundary value problems for nonlinear elliptic equations 35D30: Weak solutions

Citation

Mercaldo, Anna; Rossi, Julio D.; Segura de León, Sergio; Trombetti, Cristina. On the behaviour of solutions to the Dirichlet problem for the $p(x)$-Laplacian when $p(x)$ goes to $1$ in a subdomain. Differential Integral Equations 25 (2012), no. 1/2, 53--74. https://projecteuclid.org/euclid.die/1356012825


Export citation