On s-shaped bifurcation curves

On the exactness of an S-shaped bifurcation curve

“…A similar result was proved by S.-H. Wang [11] using the quadrature technique. In addition to obtaining an alternative proof, we do not require the boundness of f (u), as was the case in [11]. This brings up a possibility of another type of S-shaped solution curves, as in Figure 1(b).…”

“…This is an improvement of the critical constant a 0 4.4967 obtained by S.H. Wang [11]. We have thus proved the following theorem.…”

A theorem on S-shaped bifurcation curve for a positone problem with convex–concave nonlinearity and its applications to the perturbed Gelfand problem

“…The derivation of the problem and more background can refer to Bebernes and Eberly [1]. For (1.2), it has been a long-standing conjecture [3,9,10,[13][14][15]17,18] that, there exists a critical bifurcation value a 0 > 0 such that the bifurcation curve S is S-shaped for a > a 0 and is monotone increasing for 0 < a a 0 ; in particular, when a = a 0 , there is a turning point, see Fig. 1(i)-(iii).…”

“…The study of S-shaped bifurcation curve S of (1.2) and its generalization to higher dimensions has been extensively investigated by many authors, see, e.g. [3][4][5][6][7][8][9][10]14,15,18]. Brown et al [3] showed that the bifurcation curve S is S-like shaped (i.e., S has at least two turning points) for a ã ≈ 4.25 for some constantã by ap- then the bifurcation curve S is S-shaped on the (λ, u ∞ )-plane.…”

On S-shaped bifurcation curves for a class of perturbed semilinear equations

“…For all that, S-shaped bifurcation was discussed by several mathematicians in the past thirty years. The early results on exact S-shaped bifurcation are for the one-dimensional case (see [3,12,22,23]), by using the time-map method which does not work for n ≥ 2. We note that P. Korman and Y. Li [12] used some bifurcation analysis combined with time-map technique.…”