Tenth MSU Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 23 (2016), pp. 131-138.

Existence of three solutions for a two-point singular boundary-value problem with an unbounded weight

Dhanya Rajendran, Eunkyung Ko, Ratnasingham Shivaji

Abstract:
We show the existence of three solution for the singular boundary-value problem
$$\displaylines{ -z'' = h(t) \frac{f(z)}{z^\beta} \quad \text{in } (0,1) ,\cr z(t)> 0 \quad \text{in } (0,1),\cr z(0)= z(1)=0 , }$$
where $ 0 < \beta <1$, $f\in C^1([0,\infty), (0,\infty))$ and $ h\in C((0,1], (0,\infty))$ is such that $h(t)\leq C/t^\alpha$ on $(0,1]$ for some $C>0$ and $0<\alpha<1-\beta$. When there exist two pairs of sub-supersolutions $(\psi_1,\phi_1)$ and $(\psi_2,\phi_2)$ where $\psi_1\leq \psi_2\leq \phi_1, \psi_1\leq \phi_2\leq \phi_1 $ with $\psi_2 \not \leq \phi_2$, and $\psi_2 ,\phi_2$ are strict sub and super solutions. The establish the existence of at least three solutions $z_1, z_2, z_3$ satisfying $z_1\in [\psi_1,\phi_2]$, $z_2\in [\psi_2,\phi_1]$ and $z_3\in [\psi_1,\phi_1]\setminus ([\psi_1,\phi_2]\cup [\psi_2,\phi_1])$.

Published March 21, 2016.
Math Subject Classifications: 35J25, 35J55.
Key Words: Three solutions theorem; singular boundary value problems.

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  Dhanya Rajendran
Departamento de Matemática
Universidad de Concepción, Chile
email: dhanya.tr@gmail.com
Eunkyung Ko
PDE and Functional Analysis Research Center
Department of Mathematical Sciences
Seoul National University
Seoul, 151-747, South Korea
email: ekko1115@snu.ac.kr
Ratnasingham Shivaji
Department of Mathematics and Statistics
University of North Carolina at Greensboro
Greensboro, NC 27412, USA
email: shivaji@uncg.edu

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