Classification of bifurcation curves for the Minkowski-curvature problem involving general sign-changing nonlinearity

In this paper, we study the classification of bifurcation curves of positive solutions of the Minkowski-curvature problem $$ \left \{ \textstyle\begin{array}{l} -\left ( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right ) ^{\prime }= \lambda f(u),\text{ in }\left ( -L,L\right ) , \\ u(-L)=u(L)=0,\end{array}\displaystyle \right . $$ where $\lambda ,L>0$ , and $f\in C^{2}(0,\infty )$ is sign-changing. We identify and correct a significant error in (He et al. in AIMS Math. 7:17001–17018, 2022), and further refine their results. In contrast to (He et al. in AIMS Math. 7:17001–17018, 2022), which considers only the case $f(0^{+})\geq 0$ , we extend the analysis to function f satisfying the case $-\infty \leq f(0^{+})<0$ . Finally, we establish sufficient conditions for determining the exact shape of the bifurcation curve.