Bifurcation Curves in Semipositone Problems with Geometrically Concave and Concave Nonlinearities

In this paper, we study the exact multiplicity and bifurcation curves of positive solutions for the semipositone problem defined on the interval from minus one to one, with zero boundary conditions at both ends. The function f is twice continuously differentiable on the positive real line, and there exist two positive numbers such that f is positive between them and negative outside this range. We allow f at zero from the right to be negative infinity and provide many examples to illustrate these results. Furthermore, our results also yield the main theorems presented in previous references. Additionally, some earlier authors claimed to have resolved this issue under certain conditions, but we find that their proof is incorrect. Nonetheless, our results demonstrate the correctness of their conclusion.