Evolution of Positive Solution Curves in Semipositone Problems with Concave Nonlinearities1

where l )0 and f: w0,`.ªR is monotonically increasing and concave with f 0.-0 semipositone.. We establish that f should be appropriately concave by establishing conditions on f . to allow multiple positive solutions. For any l )0, we obtain the exact number of positive solutions as a function of f t.rt. We follow several families of nonlinearities f for which f9 `.[lim tª` f 9 t.)0 and study how the positive solution curves to the above problem evolve. Also, we give 1 The results in this paper were made public at the Conference on Applied Analysis of April 1996 at Louisiana State University. The corresponding paper was accepted for the proceedings of that conference published in Contemp. Math. 221 1999.. Clerical misunderstandings prevented such publication. 282 0022-247Xr00 $35.00 Copyright Q 2000 by Academic Press All rights of reproduction in any form reserved. POSITIVE SOLUTIONS OF SEMIPOSITONE PROBLEMS 283 examples where our results apply. This work extends the work of A. Castro and R. Shivaji 1988, Proc. Roy. Soc. Edinburgh Sect. A 108, 291]302.and S.-H. Wang 1994, Proc. Roy. Soc. Edinburgh 124, No. 3, 507]515.by obtaining sharper results and also gives a complete study of positive solutions for concave semipositone nonlinearities