Multivalued problem with Robin boundary condition involving diffuse measure data and variable exponent

Abstract

We study a nonlinear elliptic problem with Robin type boundary condition, governed by a general Leray–Lions operator with variable exponents and diffuse Radon measure data which does not charge the sets of zero p(·)-capacity. We prove an existence and uniqueness result of a weak solution.

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

“…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]).…”

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

“…In our paper, we consider an inhomogeneous Neumann boundary condition and diffuse Radon measure which bring some difficulty to treat. In fact, the Neumann boundary condition that appears in (P µ ) is quitely different from the one used in [19]. In order to get our main result, we define the space T 1,p(x) tr (Ω) which will help us to take into account the boundary condition.…”

Multivalued anisotropic problem with Fourier boundary condition involving diffuse Radon measure data and variable exponents

“…Note that the space in which we work is the anisotropic Sobolev space W In the classical Lebesgue and Sobolev spaces with constant exponent, many authors have studied problems with a maximal monotone graph and measure data (see [3,4,5,11,13,19]). These problems have been extended to the Sobolev spaces with variable exponent in the context of isotropic operators (see [25,27]). In this paper, we extend the study of problems with maximal monotone graph and measure data to the Sobolev spaces with variable exponents in the context of anisotropic operators.…”

“…(Ω)). To overcome this difficulty, we use the same ideas as authors in [27]. We consider a smooth domain Ω in order to work with the space W…”