Asymptotic behavior of positive solutions for the radial p-Laplacian equation

Abstract

We study the existence, uniqueness and asymptotic behavior of positive solutions to the nonlinear problem $$displaylines{ frac{1}{A}(APhi _p(u'))'+q(x)u^{alpha}=0,quad hbox{in }(0,1),cr lim_{xo 0}APhi _p(u')(x)=0,quad u(1)=0, }$$ where $alpha <p-1$, $Phi _p(t)=t|t| ^{p-2}$, A is a positive differentiable function and q is a positive measurable function in (0,1) such that for some c>0, $$ frac{1}{c}leq q(x)(1-x)^{eta }exp Big( -int_{1-x}^{eta }frac{z(s)}{s}dsBig)leq c. $$ Our arguments combine monotonicity methods with Karamata regular variation theory.

Keywords

• p-Laplacian
• asymptotic behavior
• positive solutions
• Schauder's fixed point theorem