Behavior near the origin of f′(u⁎) in radial singular extremal solutions.

Consider the semilinear elliptic equation − Δ u = λ f (u) in the unit ball B 1 ⊂ R N , with Dirichlet data u | ∂ B 1 = 0 , where λ ≥ 0 is a real parameter and f is a C 1 positive, nondecreasing and convex function in [ 0 , ∞) such that f (s) / s → ∞ as s → ∞. In this paper we study the behavior of f ′ (u ⁎) near the origin when u ⁎ , the extremal solution of the previous problem associated to λ = λ ⁎ , is singular. This answers to an open problems posed by Brezis and Vázquez [2, Open problem 5]. [ABSTRACT FROM AUTHOR]