Exhaustive existence and non-existence results for Hardy–Hénon equations in {{,mathrm{{textbf{R}}},}}^n

Abstract

This paper concerns solutions to the Hardy–Hénon equation -Δu=|x|σup\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\Delta u = |x|^\\sigma u^p \\end{aligned}$$\\end{document}in {{,mathrm{{textbf{R}}},}}^n with n ge 1 and arbitrary p, sigma in {{,mathrm{{textbf{R}}},}}. This equation was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above equation, at least one of the following three assumptions p>1,sigma ge -2, and n ge 3 is often assumed. The aim of this paper is to investigate the equation in other cases of these parameters, leading to a complete picture of the existence/non-existence results for non-trivial, non-negative solutions in the full generality of the parameters. In addition to the existence/non-existence results, the uniqueness of solutions is also discussed.