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 $$ \left\{ \begin{array}{llll} \sum\limits _{i=1}\limits^{n} D_i (a_i(x,Du(x)))=0, &x\in Ω,\\ u(x)=θ(x), & x\in \partial Ω. \end{array} \right. $$ Under some suitable conditions, they obtained an integrability result, which shows that, higher integrability of the boundary datum $θ$ forces solutions $u$ to have higher integrability as well. In the present paper, we consider ${\cal K}_{ψ,θ}^{(p_i)}$-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.