Radial solutions of p-Laplace equations with nonlinear gradient terms on exterior domains

Abstract

This paper studies the existence of radial solutions of the boundary value problem of p-Laplace equation with gradient term {−Δpu=K(|x|)f(|x|,u,|∇u|),x∈Ω,∂u∂n=0,x∈∂Ω,lim|x|→∞u(x)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\Delta_{p} u= K( \\vert x \\vert ) f( \\vert x \\vert , u, \\vert \ abla u \\vert ) ,\\quad x\\in\\Omega , \\\\ \\frac{\\partial u}{\\partial n}=0 ,\\quad x\\in\\partial\\Omega, \\\\ \\lim_{ \\vert x \\vert \ o\\infty}u(x)=0 , \\end{cases} $$\\end{document} where Ω={x∈RN:|x|>r0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega=\\{x\\in\\mathbb{R}^{N}: |x|>r_{0}\\}$\\end{document}, N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N\\ge3$\\end{document}, 1<p≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1< p\\le2$\\end{document}, K:[r0,∞)→R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$K: [r_{0}, \\infty)\ o\\mathbb{R}^{+}$\\end{document}, and f:[r0,∞)×R×R+→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f: [r_{0}, \\infty)\ imes\\mathbb{R}\ imes\\mathbb{R}^{+}\ o \\mathbb{R}$\\end{document} are continuous. Under certain inequality conditions of f, the existence results of radial solutions are obtained.