Bifurcation curves of positive solutions for the Minkowski-curvature problem with cubic nonlinearity

In this paper, we study the shape of bifurcation curve S-L of positive solutions for the Minkowski-curvature problem {-(u,(x)/root 1-(u'(x))(2))' =lambda ( -epsilon u(3) + u(2) +u+1), -L< x < L, u(-L) = u(L) =0, where lambda,epsilon > 0 are bifurcation parameters and L > 0 is an evolution parameter. We prove that there exists epsilon(0) > 0 such that the bifurcation curve S-L is monotone increasing for all L > 0 if epsilon >= epsilon(0), and the bifurcation curve S-L is from monotone increasing to S-shaped for varying L > 0 if 0 < epsilon < epsilon(0).