Renormalized solutions for some nonlinear nonhomogeneous elliptic problems with Neumann boundary conditions and right hand side measure

Abstract

Our aim in this paper is to study the existence of renormalized solution for a class of nonlinear p(x)-Laplace problems with Neumann nonhomogeneous boundary conditions and diuse Radon measure data which does not charge the sets of zero p(.)-capacity

Renormalized solutions for a p(·)-Laplacian equation with Neumann nonhomogeneous boundary condition involving diffuse measure data and variable exponent

“…where α is maximal monotone graph on R with α(0) = 0, µ is a Radon diffuse measure such that µ = µ Ω and the operator − p(x) u := −div(|∇u| p(x)−2 ∇u) is called p(•)-Laplacian, which is a natural extension of the p-Laplace operator, with p being a positive constant. The existence of renormalized solutions for the problem (P µ ) where α is a real function and µ is L 1 or Radon measure data has been studied in [3,4]. Recently in [8], problem (P µ ) was studied with µ ∈ L 1 (Ω).…”

“…Recently in [8], problem (P µ ) was studied with µ ∈ L 1 (Ω). In [3,4,8], under some assumptions the authors proved that the operator associated to the approximated problem is of type (M). So, by some a priori estimates, they obtained the convergence of the approximate sequence to a renormalized solution of the initial problem.…”

“…So, by some a priori estimates, they obtained the convergence of the approximate sequence to a renormalized solution of the initial problem. In this work, we extend the approach developed in [3,4,8] to the case of maximal monotone graph or and Radon diffuse measure data. Elliptic problems with variable exponent has been extensively studied in recent years (see [3,4,5,6,7,9,18,19,20,22,23]) and the references therein.…”

“…In this work, we extend the approach developed in [3,4,8] to the case of maximal monotone graph or and Radon diffuse measure data. Elliptic problems with variable exponent has been extensively studied in recent years (see [3,4,5,6,7,9,18,19,20,22,23]) and the references therein. The interest of studying problems with variable exponent is due to the fact they can model phenomena which arise in mathematical physics such that elastic mechanics, electro-rheological fluid dynamics and image processing (see [20,23]).…”

“…To overcome this difficulty, we use the same ideas as authors in [4,9,18,19]. We consider a suitable domain Ω in order to work in W 1,p(•) 0…”