Radial Solutions for p-k-Hessian Equations and Systems with Gradient Term

Abstract

This paper studies the existence of entire radial solutions to the p-k-Hessian equation with nonlinear gradient term σk(λDi|Du|p-2Dj(u)+α|∇u|(p-1)k=a(|x|)fk(u),x∈Rn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D u|^{p-2} D_{j}(u)\\right) \\right) +\\alpha |\ abla u|^{(p-1) k}\\right. =a(|x|) f^{k}(u), ~~x \\in \\mathbb {R}^{n}, \\end{aligned}$$\\end{document}and system with nonlinear gradient term σk(λDi|Du|p-2Dj(u)+α|∇u|(p-1)k=a(|x|)fk(v),x∈Rn,σk(λDi|Dv|p-2Dj(v)+β|∇v|(p-1)k=b(|x|)gk(u),x∈Rn.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D u|^{p-2} D_{j}(u)\\right) \\right) +\\alpha |\ abla u|^{(p-1) k}\\right. =a(|x|) f^{k}(v), ~~x \\in \\mathbb {R}^{n}, \\\\ \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D v|^{p-2} D_{j}(v)\\right) \\right) +\\beta |\ abla v|^{(p-1) k}\\right. =b(|x|) g^{k}(u), ~~x \\in \\mathbb {R}^{n}. \\end{array}\\right. \\end{aligned}$$\\end{document}By adopting monotone iteration method, we derive the existence and asymptotic behavior of the radial solutions.