Electron. J. Diff. Equ., Vol. 2012 (2012), No. 240, pp. 1-10.

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

Sonia Ben Othman, Habib Maagli

Abstract:
We study the existence, uniqueness and asymptotic behavior of positive solutions to the nonlinear problem
$$\displaylines{ \frac{1}{A}(A\Phi _p(u'))'+q(x)u^{\alpha}=0,\quad \hbox{in }(0,1),\cr \lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0, }$$
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,
$$ \frac{1}{c}\leq q(x)(1-x)^{\beta }\exp \Big( -\int_{1-x}^{\eta }\frac{z(s)}{s}ds\Big)\leq c. $$
Our arguments combine monotonicity methods with Karamata regular variation theory.

Submitted September 23, 2012. Published December 28, 2012.
Math Subject Classifications: 34B15, 35J65.
Key Words: p-Laplacian; asymptotic behavior; positive solutions; Schauder's fixed point theorem.

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Sonia Ben Othman
Département de Mathématiques
Faculté des Sciences de Tunis, Campus universitaire
2092 Tunis, Tunisia
email: Sonia.benothman@fsb.rnu.tn
Habib Mâagli
King Abdulaziz University, College of Sciences and Arts, Rabigh Campus
Department of Mathematics
P.O. Box 344, Rabigh 21911, Saudi Arabia
email: habib.maagli@fst.rnu.tn

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