Mediterr. J. Math. (2020) 17:67 https://doi.org/10.1007/s00009-020-1485-9 c  Springer Nature Switzerland AG 2020 Capacity Solution to a Nonlinear Elliptic Coupled System in Orlicz–Sobolev Spaces H. Moussa, F. Orteg´ on Gallego and M. Rhoudaf 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 = ϕ0 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 LM(Ω), M is a N -function which does not have to satisfy a Δ2 condition. Therefore, we work with generalized Orlicz– Sobolev spaces which are not necessarily reflexive. The function ϕ0 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. Mathematics Subject Classification. 35J60, 35J66, 46E30. Keywords. Coupled system, capacity solutions, nonlinear elliptic equations, weak solutions, Orlicz–Sobolev spaces. 1. Introduction This paper deals with the existence of a capacity solution to a nonlinear elliptic coupled system, whose unknowns are the temperature inside a semi- conductor material, u, and the electric potential, ϕ, namely ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − div a(x, u, ∇u)= ρ(u)|∇ϕ| 2 in Ω div(ρ(u)∇ϕ)=0 in Ω, ϕ = ϕ 0 on ∂ Ω, u =0 on ∂ Ω, (1.1) 0123456789().: V,-vol