Electron. J. Diff. Equ., Vol. 2016 (2016), No. 49, pp. 1-14.

Exact asymptotic behavior of the positive solutions for some singular Dirichlet problems on the half line

Habib Maagli, Ramzi Alsaedi, Noureddine Zeddini

Abstract:
In this article, we give an exact behavior at infinity of the unique solution to the following singular boundary value problem
$$\displaylines{ -\frac{1}{A}(Au')'=q(t)g(u), \quad t \in (0,\infty), \cr u>0, \quad \lim_{t\to 0}A(t)u'(t)=0, \quad \lim_{t\to \infty}u(t)=0. }$$
Here A is a nonnegative continuous function on $[0,\infty)$, positive and differentiable on $(0,\infty)$ such that
$$ \lim_{t\to \infty}\frac{tA'(t)}{A(t)}=\alpha>1, \quad g \in C^1((0,\infty),(0,\infty)) $$
is non-increasing on $(0,\infty)$ with $\lim_{t\to 0}g'(t)\int_0^t\frac{ds}{g(s)}=-C_g\leq 0$ and the function q is a nonnegative continuous, satisfying
$$ 0<a_1=\liminf_{t\to \infty}\frac{q(t)}{h(t)} \leq \limsup_{t\to \infty}\frac{q(t)}{h(t)}=a_2<\infty, $$
where $h(t)=c t^{-\lambda}\exp (\int_{1}^{t }\frac{y(s)}{s}ds)$, $\lambda \geq 2$, $c>0$ and y is continuous on $[ 1,\infty)$ such that $\lim_{t\to \infty}y(t)=0$.

Submitted December 8, 2015. Published February 17, 2016.
Math Subject Classifications: 34B16, 34B18, 34D05.
Key Words: Singular nonlinear boundary value problems; positive solution; exact asymptotic behavior; Karamata regular variation theory.

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Habib Mâagli
Department of Mathematics, College of Sciences and Arts
King Abdulaziz University, Rabigh Campus
P.O. Box 344, Rabigh 21911, Saudi Arabia
email: habib.maagli@fst.rnu.tn
Ramzi Alsaedi
Department of Mathematics
Faculty of Sciences, King Abdulaziz University
P.O. Box 80203, Jeddah 21589, Saudi Arabia
email: ramzialsaedi@yahoo.co.uk
Noureddine Zeddini
Department of Mathematics, College of Sciences and Arts
King Abdulaziz University, Rabigh Campus
P.O. Box 344, Rabigh 21911, Saudi Arabia
email: noureddine.zeddini@ipein.rnu.tn

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