On solutions of anisotropic elliptic equations with variable exponent and measure data

Abstract

Anisotropic elliptic equations of the second order with variable exponents in nonlinearities and the right-hand side as a diffuse measure are considered in the space R n . The existence of an entropy solution in anisotropic Sobolev-Orlicz spaces with variable exponents is proved. The obtained entropy solution is shown to be a renormalized solution. (2000). Primary 35J62; Secondary 35J25. Mathematics Subject Classification

Entropy solution for an equation with measure-valued potential in a hyperbolic space

“…In unbounded domains with infinite measure and N -function M (x, u) = |u| p(x) the existence of an entropy solution and a renormalized solution of the equation − div(a(x, u, ∇u)) + b(x, u, ∇u) + |u| p(x)−1 = σ was established for the first time in [7] and [8], for measures σ of a special form.…”

Anisotropic problem with non-local boundary conditions and measure data

“…For example, at any point in Ω the partial differential equation and/or the boundary conditions may contains integrals of the unknown u over parts of Ω, values of u elsewhere in D or, generally speaking, some non-local operator on u. Beside the mathematical interest of nonlocal conditions, it seems that this type of boundary condition appears in petroleum engineering model for well modeling in a 3D stratified petroleum reservoir with arbitrary geometry (see [12] and [15]). A lot of papers ( see [34], [24], [25], [2], [19], [1]) on problems like (1.1) considered cases of generally boundary value condition. In [6], Bonzi et al studied the following problems.…”