Behavior near the origin of $f^{\prime }(u^{\ast })$ in radial singular extremal solutions

Abstract

Consider the semilinear elliptic equation $-\Delta u=\lambda f(u)$ in the unit ball $B_1\subset {\mathbb{R}}^N$, with Dirichlet data $u{|}_{\partial B_1}=0$, where $\lambda \ge 0$ is a real parameter and $f$ is a $C^1$ positive, nondecreasing and convex function in $[0,\infty )$ such that $f(s)/s\to \infty$ as $s\to \infty$. In this paper we study the behavior of $f^{\prime }(u^{\ast })$ near the origin when $u^{\ast }$, the extremal solution of the previous problem associated to $\lambda ={\lambda }^{\ast }$, is singular. This answers to an open problems posed by Brezis and Vázquez.

arXiv comment14 pages