In this paper, we show the positive solutions set for one-dimensional $p$-Laplacian problem with sign-changing weight contains a reversed $S$-shaped continuum. By figuring the shape of unbounded continuum of positive solutions, we identify the interval of bifurcation parameter in which the $p$-Laplacian problem has one or two or three positive solutions according to the asymptotic behavior of nonlinear term at 0 and $\mathrm{\infty }$. The proof of the main result is based upon bifurcation technique.

In this paper, we study the shape of the bifurcation curves of positive solutions for the one-dimensional Minkowski-curvature problem. By developing some new time mapping techniques, we find that the bifurcation curve is $\subset$-shaped/monotone increasing/ $S$-like shaped on the $(\lambda ,||u|{|}_{\mathrm{\infty }})$ plane when the nonlinearity satisfies different assumptions. Finally, two examples are given to illustrate our result.