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

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 $$ 00$ and y is continuous on $[ 1,\infty)$ such that $\lim_{t\to \infty}y(t)=0$.

Keywords

• Singular nonlinear boundary value problems
• positive solution
• exact asymptotic behavior
• Karamata regular variation theory