On some nonlinear elliptic equations with measurable boundary conditions in anisotropic weighted Sobolev spaces

Abstract

The novelty of this paper is to establish an existence of entropy solutions for anisotropic elliptic equation looks like -div B(x, ν, ∇ ν)+H(x, ν)=f-div F where the datum f ∈ L¹(Ω) and F ∈ ∏_i=1^N L^p′_i(Ω, w_i*) and H(x, ν) ∈ L¹(Ω). Furthermore, only the large monotonicity conditions will be assumed on B(x, s, ξ). To overcome this difficulty, we will use the approach of Minty's lemma in the anisotropic weighted Sobolev spaces.