Bifurcation Curves in Semipositone Problems with Geometrically Concave and Concave Nonlinearities

TL;DR

This paper analyzes the exact multiplicity and bifurcation structure of positive solutions to the semipositone boundary value problem $-u''=\lambda f(u)$ on $(-1,1)$ with zero boundary data, allowing sign-changing nonlinearities with $F(u)=\int_0^u f(t)\,dt$ and $F(\eta)=0$. It develops a time-map approach, linking the bifurcation curve $S$ to a parametric family $T(\alpha)$ and deriving precise endpoint behavior and shape (monotone vs $\subset$-shaped) under structural conditions (H$_1$)-(H$_4$) and a sign criterion $G<0$. The paper provides a comprehensive collection of examples illustrating how different $f$–growth patterns yield different bifurcation shapes, and it clarifies gaps in earlier proofs. These results generalize prior theorems for convex/concave nonlinearities and yield explicit criteria for the exact multiplicity of positive solutions. An Appendix discusses a subtle flaw in a previous proof and reinforces the correctness of the present time-map framework.

Abstract

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.

Figures (7)

• Figure 1: The graph of $f$. (i) $\beta _{2}<\infty$. (ii) $\beta _{2}=\infty$.
• Figure 2: The graphs of the bifurcation curve $S$ of (\ref{['eq1']}) with $\beta _{2}=\infty$ if (\ref{['h1']}) holds.
• Figure 3: Graphs of bifurcation curves $S$ of (\ref{['eq1']}) with $f(0^{+})>-\infty$.
• Figure 4: Graphs of the bifurcation curve $S$ if $f^{\prime \prime }(u)>0$ on $\left( 0,\infty \right)$. (i) $f(0^{+})=0$ and $\lim\limits_{u\rightarrow \infty }f^{\prime }(u)=\infty$. (ii) $f(0^{+})=0$ and $\lim\limits_{u\rightarrow \infty }f^{\prime }(u)\in (0,\infty )$. (iii) $f(0^{+})<0$ and $\lim\limits_{u\rightarrow \infty }f^{\prime }(u)=\infty$. (iv) $f(0^{+})<0$ and $\lim\limits_{u\rightarrow \infty }f^{\prime }(u)\in (0,\infty ).$
• Figure 5: (i) The graph of $-2u^{3}+3u^{2}+3$ on $\left( 0,3\right)$. (ii) The graph of $-3u^{4}-4u^{3}+30u^{2}-36u-3$ on $\left( 1,3\right) .$

Theorems & Definitions (31)

• Theorem 1: ref10
• Theorem 2: ref8
• Theorem 3
• Remark 1
• Remark 2
• Theorem 4
• Remark 3
• Corollary 1
• Theorem 5
• Corollary 2