Electron. J. Diff. Equ., Vol. 2015 (2015), No. 235, pp. 1-12.

Combined effects in nonlinear singular second-order differential equations on the half-line

Imed Bachar

Abstract:
We consider the existence, uniqueness and the asymptotic behavior of positive continuous solutions to the second-order boundary-value problem
$$\displaylines{ \frac{1}{A}(Au')'+a_1(t)u^{\sigma _1}+a_2(t)u^{\sigma _2}=0, \quad t\in (0,\infty ), \cr \lim_{t\to 0^+} u(t)=0, \quad \lim_{t\to \infty } \frac{u(t)}{\rho (t)}=0, }$$
where $\sigma _1,\sigma _2\in (-1,1)$, A is a continuous function on $[0,\infty )$, positive and differentiable on $(0,\infty )$ such that $\int_0^1\frac{1}{A(t)}dt<\infty $ and $\int_0^{\infty }\frac{1}{A(t)}dt=\infty $. Here $\rho (t)=\int_0^{t}\frac{1}{A(s)}ds$ and for $i\in \{1,2\}$, $a_i$ is a nonnegative continuous function in $(0,\infty )$ such that there exists c%>0 satisfying for t>0,
$$ \frac{1}{c}\frac{h_i(m(t))}{A^{2}(t)( 1+\rho (t)) ^{\mu _i}} \leq a_i(t) \leq c\frac{h_i(m(t))}{A^{2}(t) ( 1+\rho (t)) ^{\mu_i}}, $$
where $m(t)=\frac{\rho (t)}{1+\rho (t)}$ and $h_i(t)=c_it^{- \lambda _i}\exp (\int_{t}^{\eta }\frac{z_i(s)}{s}ds)$, $c_i>0$, $\lambda _i\leq 2$, $\mu _i>2$ and $z_i$ is continuous on $[0,\eta ]$ for some $\eta >1$ such that $z_i(0)=0$. The comparable asymptotic rate of $a_i(t)$ determines the asymptotic behavior of the solution.

Submitted May 3, 2015. Published September 11, 2015.
Math Subject Classifications: 34B15, 34B18, 34B27.
Key Words: Green's function; Karamata regular variation theory; positive solution; Schauder fixed point theorem.

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Imed Bachar
King Saud University College of Science
Mathematics Department, P.O. Box 2455
Riyadh 11451, Saudi Arabia
email: abachar@ksu.edu.sa

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