Subsolutions: A Journey from Positone to infinite Semipositone Problems

Start: 02/13/2013 - 4:15pm
End  : 02/13/2013 - 5:15pm


Ratnasingham Shivaji, University of North Carolina at Greensboro


We discuss the existence of positive solutions to −∆u = λf (u) in Ω, with u = 0 on the boundary, where λ is a positive parameter, Ω is a bounded domain with smooth boundary, ∆ is the Laplacian operator, and f : (0, ∞) → R is a continuous function. We first discuss the cases when f (0) > 0 (positone) 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 ≫ 1 (which will lead to large supersolutions) we discuss the existence of positive solutions. Finally, we will discuss the case of infinite semipositone problems (lims→0+ f (s) = −∞).

Beckman B126, Harvey Mudd College

Shivaji.pdf139.86 KB

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