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

We consider the existence, uniqueness and the asymptotic behavior of positive continuous solutions to the second-order boundary-value problem
 \frac{1}{A}(Au')'+a_1(t)u^{\sigma _1}+a_2(t)u^{\sigma _2}=0,
 \quad t\in (0,\infty ), 
 \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.

Show me the PDF file (255 KB), TEX file, and other files for this article.

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

Return to the EJDE web page