Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Minkowski-curvature problem

$\{\begin{array}{c}?{(u^{\prime }/\sqrt{1?u^{\prime 2}})}^{\prime }=\lambda f(u),\text{ in }(?L,L),\hfill \\ u(?L)=u(L)=0,\hfill \end{array}$

where $\lambda ,L>0$, $f\in C[0,\infty )\cap C^2(0,\infty )$ and $f(u)>0$ for $u\ge 0$. Furthermore, we show that, for sufficiently large $L>0$, the bifurcation curve $S_L$ may have arbitrarily many turning points. Finally, we apply these results to obtain the global bifurcation diagrams for Ambrosetti–Brezis–Cerami problem, Liouville–Bratu–Gelfand problem and perturbed Gelfand problem with the Minkowski-curvature operator, respectively. Moreover, we will make two lists which show the different properties of bifurcation curves for Minkowski-curvature problems, corresponding semilinear problems and corresponding prescribed curvature problems.