Capacity Solution to a Nonlinear Elliptic Coupled System in Orlicz–Sobolev Spaces

Abstract

We analyze the existence of a capacity solution to the following nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, ϕ, −Au = ρ(u)|∇ϕ|2 in Ω, div(ρ(u)∇ϕ) = 0 in Ω, ϕ = ϕ_0 on ∂Ω, u = 0 on∂Ω, where Ω ⊂ R^d, d ≥ 2 and Au = −div a(x, u,∇u) is a Leray–Lions operator defined on W^1_0 L_M(Ω), M is a N-function which does not have to satisfy a Δ_2 condition. Therefore, we work with generalized Orlicz– Sobolev spaces which are not necessarily reflexive. The function ϕ_0 is given. The proof combines truncation methods, monotonicity techniques and regularizing methods in Orlicz spaces. We introduce a sequence of approximate problems which converges (up to a subsequence) in a certain sense to a capacity solution in the context of non-reflexive Orlicz–Sobolev spaces.