Remarks on a paper by Leonetti and Siepe

In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type {∑i=1nDi(ai(x,Du(x)))=0,x∈Ω,u(x)=θ(x),x∈∂Ω. Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum θθ forces solutions uu to have higher integrability as well. In the present paper, we consider Kψ,θ(pi)-obstacle problems of the nonhomogeneous anisotropic elliptic equations ∑i=1nDi(ai(x,Du(x)))=∑i=1nDifi(x) under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe.