Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data

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

“…For the corresponding isotropic parabolic equations with L 1 -data, existence and uniqueness of renormalized solutions is established in Blanchard and Murat [22], see also Rakotoson [38,39] and Lions [37] for some time dependent problems motivated by the Navier-Stokes equations. Finally, let us mention that an equivalent notion of solutions, called entropy solutions, was introduced independently by Bénilan et al [15], see also [11], several researches deals with the existence solutions of elliptic and parabolic problems under various assumptions and in different contexts (see [42,32,43,6,8,19,1,7,5,2,16,9,17,20,3,10,4,18,30,31,29,25,21]).…”

Existence of entropy solutions for degenerate elliptic unilateral problems with variable exponents

“…Otherwise, the term H(x, u, ∇u) is said to be an absorption term, in this case a detailed picture of what happens is available (see e.g. [4,6,8,9,10,11]). …”

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

“…Observe that for elliptic equations with boundary Dirichlet conditions and L 1 -data, this notion is equivalent to the notion of entropy solutions (see [1,2,3,18]) and to the notion of solutions obtained as limit of approximations. As far as elliptic equations with L 1 -data and Dirichlet boundary conditions are concerned we refer to [1,2,3,7,18] among a wide literature.…”