The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems

We study the Dirichlet boundary value problem $\left\{\begin{array}{c}{u}^{″}\left(t\right)+\lambda f\left(u\left(t\right)\right)=0,-1<t<1,\\ u\left(-1\right)=u\left(1\right)=0,\end{array}\right.$ generally and develop a schema for determining the relationship between the values of its parameters and the number of positive solutions. Then, we focus our attention on the special cases when $f\left(u\right)=\left(\sigma -u\right)\exp \left(-K/\left(1+u\right)\right)$ and $f\left(u\right)={\displaystyle {\prod }_{i=1}^m\left(a_i—u\right)}$ , respectively. We prove first that all positive solutions of the first problem are less than or equal to $\sigma$ , obtain more specific lower and upper bounds for these solutions, and compute a curve in the $\sigma K$ -plane with accuracy up to ${10}^{-6}$ , below which the first problem has a unique positive solution and above which it has exactly three positive solutions. For the second problem, we determine its number of positive solutions and find a formula for the value of $\lambda$ that separates the regions of $\lambda$ , in which the problem has different numbers of solutions. We also computed the graphs for some special cases of the second problem, and the results are consistent with the existing results. Our code in Mathematica is available upon request.