Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

For 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p < \infty $$\end{document}, we prove radial symmetry for bounded nonnegative solutions of -divw(x)H(∇u)p-1∇ξH(∇u)=f(u)w(x)inΣ∩Ω,u=0onΓ0,⟨∇ξH(∇u),ν⟩=0onΓ1\0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathop {\mathrm {div}}\left\{ w(x) \, H( \nabla u )^{p-1}\, \nabla _{ \xi } H( \nabla u) \right\} = f(u) \, w(x) \, &{} \text{ in } \ \Sigma \cap \Omega , \\ u=0 \, &{} \text{ on } \ \Gamma _0 , \\ \langle \nabla _\xi H(\nabla u) , \nu \rangle = 0 \, &{} \text{ on } \ \Gamma _1 {\setminus } \left\{ 0 \right\} , \end{array}\right. } \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a Wulff ball, Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} is a convex cone with vertex at the center of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, Γ0:=Σ∩∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _0 := \Sigma \cap \partial \Omega $$\end{document}, Γ1:=∂Σ∩Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _1 := \partial \Sigma \cap \Omega $$\end{document}, H is a norm, w is a given weight and f is a possibly discontinuous nonnegative nonlinearity.