Structural stability for variable exponent elliptic problems, II: The p(u)p(u)-Laplacian and coupled problems

We study well-posedness for elliptic problems under the form b(u)−diva(x,u,∇u)=f, where aa satisfies the classical Leray–Lions assumptions with an exponent pp that may depend both on the space variable xx and on the unknown solution uu. 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 NN. Then, under mild regularity assumptions on ΩΩ and on the nonlinearities, we show that the associated solution operator is an order-preserving contraction in L1(Ω)L1(Ω).In addition, existence analysis for a sample coupled system for unknowns (u,v)(u,v) involving the p(v)p(v)-Laplacian of uu is carried out. Coupled elliptic systems with similar structure appear in applications, e.g. in modelling of stationary thermorheological fluids.