Homogenization of a nonlinear elliptic system with a transport term in L 2

We consider the homogenization of a non-linear elliptic system of two equations related to some models in chemotaxis and flows in porous media. One of the equations contains a convection term where the transport vector is only in L 2 and a right-hand side which is only in L 1 . This makes it necessary to deal with entropy or renormalized solutions. The existence of solutions for this system has been proved in reference (Comm. Partial Differential Equations 45(7) (2020) 690–713). Here, we prove its stability by homogenization and that the correctors corresponding to the linear diffusion terms still provide a corrector for the solutions of the non-linear system.

Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L 1-data

We prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.

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

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

Existence of renormalized solutions for some quasilinear elliptic Neumann problems

This paper is devoted to study some nonlinear elliptic Neumann equations of the type { A u + g ( x , u , ∇ u ) + | u | q ( ⋅ ) - 2 u = f ( x , u , ∇ u ) in Ω , ∑ i = 1 N a i ( x , u , ∇ u ) ⋅ n i = 0 on ∂ Ω , \left\{ {\matrix{ {Au + g(x,u,\nabla u) + |u{|^{q( \cdot ) - 2}}u = f(x,u,\nabla u)} \hfill & {{\rm{in}}} \hfill & {\Omega ,} \hfill \cr {\sum\limits_{i = 1}^N {{a_i}(x,u,\nabla u) \cdot {n_i} = 0} } \hfill & {{\rm{on}}} \hfill & {\partial \Omega ,} \hfill \cr } } \right. in the anisotropic variable exponent Sobolev spaces, where A is a Leray-Lions operator and g(x, u, ∇u), f (x, u, ∇u) are two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem.