Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 123-131.

Subsolutions: A journey from positone to infinite semipositone problems

Eun Kyoung Lee, Ratnasingham Shivaji, Jinglong Ye

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.

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Eun Kyoung Lee
Department of Mathematics and Statistics, Center for Computational Sciences
Mississippi State University, Mississippi State, MS 39762, USA
email: el165@msstate.edu
Ratnasingham Shivaji
Department of Mathematics and Statistics, Center for Computational Sciences
Mississippi State University, Mississippi State, MS 39762, USA
email: shivaji@ra.msstate.edu
Jinglong Ye
Department of Mathematics and Statistics, Mississippi State University
Mississippi State, MS 39762, USA
email: jy79@msstate.edu

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