Evolution of positive solution curves in semipositone problems with concave nonlinearities

We study the existence, multiplicity, and stability of positive solutions to -u "(x) = lambda f(u(x)) for x is an element of (-1,1). u(-1) = 0 = u(1), where lambda > 0 and f: [0,infinity) --> R 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 lambda > 0, we obtain the exact number of positive solutions as a function of f(t)/t. We follow several families of nonlinearities f for which f'(infinity) := lim(t-->infinity) f'(t) > 0 and study how the positive solution curves to the above problem evolve. Also, we give 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. Sec. Edinburgh 124, No. 3, 507-515) by obtaining sharper results and also gives a complete study of positive solutions for concave semipositone nonlinearities. (C) 2000 Academic Press.