In the recent past many results have been established on non-negative solutions to boundary value problems of the form
-u''(x) = λf(u(x)); 0 < x < 1,
u(0) = 0 = u(1)
where λ>0, f(0)>0 (positone problems). In this paper we consider the impact on the non-negative solutions when f(0)<0. We find that we need f(u) to be convex to guarantee uniqueness of positive solutions, and f(u) to be appropriately concave 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.
© 1988 Royal Society of Edinburgh
A. Castro and R. Shivaji. “Non-negative solutions for a class of non-positone problems,” Proc. Royal Soc. Edinburgh, 108A, (1988), pp. 291-302.