Abstract: We
study the positive solutions to the two point
boundary value problem
where
l > 0is
a parameter and fÎC2[0, ¥)is
monotonically increasing with f(0) < 0(semipositone).
We find that we need f(u)to
be convex to guarantee uniqueness of positive
solutions, and f(u)to
be appropriately convex for multiple positive
solutions. This is in contrast to the case of
positone problems, where the roles of
convexity and concavity were interchanged to
obtain similar results. We further establish
the existence of non-negative solutions with
interior zeros, which did not exist in
positone problems. Also, we obtain the exact
number of positive solutions as a function of f(t)/t.
Keywords and phrases: semipositone problems, concave nonlinearities, non-negative solutions.
