Electron. J. Differential Equations, Vol. 2018 (2018), No. 193, pp. 1-13.

Infinite semipositone problems with a falling zero and nonlinear boundary conditions

Mohan Mallick, Lakshmi Sankar, Ratnasingham Shivaji, Subbiah Sundar

We consider the problem
 -u'' =h(t)\big(\frac{au-u^{2}-c}{u^\alpha}\big) , \quad t \in (0, 1),\cr
 u(0) = 0, \quad u'(1)+g(u(1))=0,
where $a>0$, $c\geq 0$, $\alpha \in (0, 1)$, $h{:}(0, 1] \to (0, \infty)$ is a continuous function which may be singular at $t=0$, but belongs to $L^1(0, 1)\cap C^1(0,1)$, and $g{:}[0, \infty) \to [0, \infty)$ is a continuous function. We discuss existence, uniqueness, and non existence results for positive solutions for certain values of a, b and c.

Submitted October 15, 2018. Published November 27, 2018.
Math Subject Classifications: 35J25, 35J66. 35J75.
Key Words: Infinite semipostione; exterior domain; sub and super solutions; nonlinear boundary conditions.

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Mohan Mallick
Department of Mathematics
IIT Madras, Chennai-600036, India
email: mohan.math09@gmail.com
Lakshmi Sankar
Department of Mathematics
IIT Palakkad, Kerala-678557, India
email: lakshmi@iitpkd.ac.in
Ratnasingham Shivaji
Department of Mathematics and Statistics
University of North Carolina at
Greensboro, NC 27412, USA
email: shivaji@uncg.edu
Subbiah Sundar
Department of Mathematics
IIT Madras, Chennai-600036, India
email: slnt@iitm.ac.in

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