Integrability for solutions to some anisotropic elliptic equations

We consider the boundary value problem {∑i=1nDi(ai(x,Du(x)))=0,x∈Ω;u(x)=u∗(x),x∈∂Ω. We show that, higher integrability of the boundary datum u∗u∗ forces solutions uu to have higher integrability as well. Assumptions on ai(x,z)ai(x,z) are suggested by the Euler equation of the anisotropic functional ∫Ω(|D1u|p1+|D2u|p2+⋯+|Dnu|pn)dx.∫Ω(|D1u|p1+|D2u|p2+⋯+|Dnu|pn)dx.