Asymptotic Behavior of Positive Solutions of a Nonlinear Combined p-Laplacian Equation

For p > 1, we establish existence and asymptotic behavior of a positive continuous solution to the following boundary value problem 1AAΦp(u′)′+a1(r)uα1+a2(r)uα2=0, in(0,∞),limr→0AΦp(u′)(r)=0,limr→∞u(r)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{ll}\frac{1}{A} \left( A\Phi _{p}(u^{\prime})\right) ^{\prime}+a_{1}(r)u^{\alpha _{1}}+a_{2}(r)u^{\alpha _{2}}=0, \, {\rm in}\, (0,\infty ),\\ {\rm lim}_{r\rightarrow 0} A\Phi _{p}(u^{\prime})(r)=0, {\rm lim}_{r\rightarrow \infty } u(r)=0,\end{array}\right.$$\end{document}where α1,α2<p-1,Φp(t)=t|t|p-2,A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha _{1}, \alpha _{2} < p -1, \Phi _{p}(t) = t|t| ^{p-2},A}$$\end{document} is a positive differentiable function and a1, a2 are two positive functions in Clocγ((0,∞)),0<γ<1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C_{\rm loc}^{\gamma}((0, \infty )), 0 < \gamma < 1,}$$\end{document} satisfying some appropriate assumptions related to Karamata regular variation theory. Also, we obtain an uniqueness result when α1, α2∈(1-p,p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha _{1}, \alpha _{2} \in (1-p,p-1)}$$\end{document}. Our arguments combine a method of sub and supersolutions with Karamata regular variation theory.