Seventh Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 17 (2009), pp. 123-131. 

Title: Subsolutions: A journey from positone to infinite semipositone problems

Authors: Eun Kyoung Lee       (Mississippi State Univ., MS, USA)
         Ratnasingham Shivaji (Mississippi State Univ., MS, USA)
	 Jinglong Ye          (Mississippi State Univ., MS, USA)
 
Abstract: 
 We discuss the existence of positive solutions to
 $-\Delta u=\lambda f(u)$ in $\Omega$, with $u=0$ on the boundary, where
 $\lambda$ is a positive parameter, $\Omega$ is a bounded domain with
 smooth boundary $\Delta $ is the Laplacian operator, and
 $f:(0,\infty)\to R$ is a continuous function. We first
 discuss the cases when $f(0)>0$ (positone), $f(0)=0$ and $f(0)<0$
 (semipositone). In particular, we will review the existence of
 non-negative strict subsolutions. Along with these subsolutions and
 appropriate assumptions on $f(s)$ for $s\gg 1$ (which will lead to
 large supersolutions) we discuss the existence of positive
 solutions. Finally, we obtain new results on the case of infinite
 semipositone problems ($\lim_{s\to 0^{+}}f(s)=-\infty$).

Published April 15, 2009.
Math Subject Classifications: 35J25.
Key Words: Positone; semipositone; infinite semipositone; 
           sub-super solutions.