Positive solutions for a class of infinite semipositone problems

Abstract

We analyze the positive solutions to the singular boundary value problem \[-\Delta u = \lambda[ f(u)-1/u^{\alpha}]; x \in \Omega\] \[u = 0; \, x \in \partial\Omega,\] where $f$ is a $C^2$ function in $(0,\infty)$, $f(0)\geq 0,f^{'}>0, \lim_{s\rightarrow\infty}\frac{f(s)}{s}=0, \lambda$ is a positive parameter, $\alpha \in (0,1)$ and $\Omega$ is a bounded region in $ R^{n}, n \geq 1 $ with $C^{2+\gamma}$ boundary for some $\gamma \in (0,1)$. In the case $n=1$ we use the quadrature method and for $n>1$ we use the method of sub-super solution to establish our results.