Infinite semipositone problems with a falling zero and nonlinear boundary conditions

Abstract

We consider the problem $$\displaylines{ -u'' =h(t)\big(\frac{au-u^{2}-c}{u^\alpha}\big) , \quad t \in (0, 1),\cr u(0) = 0, \quad u'(1)+g(u(1))=0, }$$ 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.

Keywords

• Infinite semipostione
• exterior domain
• sub and super solutions
• nonlinear boundary conditions