Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R^{n}backslash{0}

Imed Bachar,Entesar Aljarallah

doi:10.1186/s13661-022-01584-3

Abstract

We consider the following singular semilinear problem {Δu(x)+p(x)uγ=0,x∈D(in the distributional sense),u>0,in D,lim|x|→0|x|n−2u(x)=0,lim|x|→∞u(x)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} \\Delta u(x)+p(x)u^{\\gamma }=0,\\quad x\\in D ~(\\text{in the distributional sense}), \\\\ u>0,\\quad \\text{in }D, \\\\ \\lim_{ \\vert x \\vert \\rightarrow 0} \\vert x \\vert ^{n-2}u(x)=0, \\\\ \\lim_{ \\vert x \\vert \\rightarrow \\infty }u(x)=0,\\end{cases} $$\\end{document} where gamma <1, D=mathbb{R}^{n}backslash {0} (ngeq 3) and p is a positive continuous function in D, which may be singular at x=0. Under sufficient conditions for the weighted function p(x), we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

Highlights

Introduction and the main resultSemilinear elliptic partial differential equations of the type u(x) + p(x)uγ = 0 (1.1)will be considered in D = Rn\{0} (n ≥ 3), where γ < 1 and p is a positive continuous function in D, which may be singular at x = 0
Under sufficient conditions for the weighted function p(x), we prove the existence of a positive continuous solution on D, which could blow-up at the origin
In [5], by using Karamata regular variation theory and the sub-supersolutions method, the authors studied the asymptotic behavior as |x| → ∞ of the unique classical positive solution of problem (1.3) with γ < 1 and p(x) is a nonnegative function in Clαoc(Rn), 0 < α

Summary

Introduction

Under sufficient conditions for the weighted function p(x), we prove the existence of a positive continuous solution on D, which could blow-up at the origin. In [20], the authors proved the existence and uniqueness of a positive continuous solution to the nonlinear elliptic problem In [5], by using Karamata regular variation theory and the sub-supersolutions method, the authors studied the asymptotic behavior as |x| → ∞ of the unique classical positive solution of problem (1.3) with γ < 1 and p(x) is a nonnegative function in Clαoc(Rn), 0 < α

Results

Conclusion