Remarks on a Paper by Leonetti and Siepe

Abstract

In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type $$\{\begin{array}{ll}\sum _{i=1}^n{D}_i(a_i(x,Du(x)))=0,& x\in \Omega ,\\ u(x)=\theta (x),& x\in \partial \Omega .\end{array}$$ Under some suitable conditions, they obtained an integrability result, which shows that, higher integrability of the boundary datum $\theta$ forces solutions $u$ to have higher integrability as well. In the present paper, we consider ${\mathcal{K}}_{\psi ,\theta }^{(p_i)}$-obstacle problems of the nonhomogeneous anisotropic elliptic equations $$\sum _{i=1}^n{D}_i(a_i(x,Du(x)))=\sum _{i=1}^n{D}_i{f}^i(x).$$ 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.