We study well-posedness for elliptic problems under the form
b(u)−div 𝔞(x,u,∇u)=f,
where 𝔞 satisfies the classical Leray-Lions assumptions with an exponent p that may depend both on the space variable x and on the unknown solution u. A prototype case is the equation u−div (|∇u|p(u)−2∇u)=f.
We have to assume that infx∈Ω⎯⎯⎯⎯⎯,z∈ℝp(x,z) is greater than the space dimension N. Then, under mild regularity assumptions on Ω and on the nonlinearities, we show that the associated solution operator is an order-preserving contraction in L1(Ω).
In addition, existence analysis for a sample coupled system for unknowns (u,v) involving the p(v)-Laplacian of u is carried out. Coupled elliptic systems with similar structure appear in applications, e.g., in modelling of stationary thermorheological fluids.

variable exponent; p(u)-Laplacian; thermorheological fluids; well-posedness; Young measures