Abstract
We study the following system of equations: \t\t\t0.1{Δu1=p1(|x|)f1(u1,u2)in RN,Δu2=p2(|x|)f2(u1,u2)in RN.\\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}$$ \\textstyle\\begin{cases} \\Delta u_{1}=p_{1} ( \\vert x \\vert ) f_{1} ( u_{1},u_{2} ) &\\text{in }\\mathbb{R}^{N}, \\\\ \\Delta u_{2}=p_{2} ( \\vert x \\vert ) f_{2} ( u_{1},u_{2} )& \\text{in }\\mathbb{R}^{N}. \\end{cases} $$\\end{document} Here f_{1}, f_{2} are continuous nonlinear functions that satisfy Keller–Osserman-type conditions, and p_{1} and p_{2} are continuous weight functions. We establish the existence of radial solutions for this system under various boundary conditions.
Highlights
1 Introduction In this paper, we study the existence and asymptotic behavior of positive radial solutions of the following semilinear elliptic system:
We study system (1.1) under three different sets of boundary conditions:
For clarity and ease of presentation, we introduce the following notations: r = |x| and a, b, c1, c2, ε1, ε2 ∈ (0, ∞) are suitably chosen, r t
Summary
1 Introduction In this paper, we study the existence and asymptotic behavior of positive radial solutions of the following semilinear elliptic system: In the particular case of R = ∞ and p1(x) = p2(x) = 1, Lair and Mohammed [10] proved that system (1.6) has a positive entire large radial solution if and only if 0 ≤ max{α, ν} ≤ 1 and βγ ≤ (1 – α)(1 – ν).
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