Regularity for solutions to anisotropic obstacle problems

For Ω a bounded subset of ℝn, n ⩾ 2, ψ any function in Ω with values in ℝ ∪ {±∞} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in W^{1,\left( {q_i } \right)} \left( \Omega \right)$$\end{document}, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{K}_{\psi ,\theta }^{\left( {q_i } \right)} \left( \Omega \right) = \left\{ {v \in W^{1,\left( {q_i } \right)} \left( \Omega \right):v \geqslant \psi , a.e. and v - \theta \in W_0^{1,\left( {q_i } \right)} \left( \Omega \right)} \right\}.$$\end{document} This paper deals with solutions to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{K}_{\psi ,\theta }^{\left( {q_i } \right)}$$\end{document}-obstacle problems for the A-harmonic equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- div\mathcal{A}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right) = - div f\left( x \right)$$\end{document} as well as the integral functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\left( {u;\Omega } \right) = \int_\Omega {f\left( {x,u\left( x \right),\nabla u\left( x \right)} \right)dx.}$$\end{document} Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.