A bifurcation diagram of solutions to semilinear elliptic equations with general supercritical growth.

We study the global bifurcation diagram of the positive solutions to the problem { Δ u + λ f (u) = 0 in B , u = 0 on ∂ B , where B is the unit ball in R N with N ≥ 3. Under general supercritical growth conditions on f (u) , we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f (u) , and we exhibit that a bifurcation curve for Δ u + λ f (u) = 0 has the same qualitative property as a classical Gel'fand problem Δ u + λ e u = 0 for N ≥ 3 except N = 10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. [ABSTRACT FROM AUTHOR]