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Bulletin of the London Mathematical Society 2006 38(6):1033-1044; doi:10.1112/S0024609306018984
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© Oxford University Press

Uniqueness and Nonexistence of Positive Solutions to Semipositone Problems

E. Norman Dancer and Junping Shi

School of Mathematics and Statistics, University of Sydney Sydney, NSW2006, Australia e-mail: normd{at}maths.usyd.edu.au
Department of Mathematics, College of William and Mary Williamsburg, VA 23187, USA and School of Mathematics, Harbin Normal University Harbin, Heilongjiang 150080, PRC e-mail: shij{at}math.wm.edu

Received 23 June 2005. Revision received 2 December 2005.

We consider the uniqueness of the positive solution to a semilinear elliptic equation with Dirichlet boundary condition and the nonlinearity satisfying f(0)<0 and having asymptotic sublinear growth rate. A similar idea is also applied to the nonexistence of a positive solution to a superlinear problem. 2000 Mathematics Subject Classification 35J65 (primary), 35B40 (secondary).

The first author is partially supported by the Australia Research Council; the second author is partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary junior research leave, and an Overseas Scholar grant from Heilongjiang Province, China.


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