Behavior near the origin of f'(u^∗) in radial singular extremal solutions

2020/05/28 by Villegas, Salvador
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.14334

Abstract

Consider the semilinear elliptic equation -Δu=λf(u) in the unit ball B₁⊂ ℝ^N, with Dirichlet data u|_{∂ B1}=0, where λ≥ 0 is a real parameter and f is a C¹ 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.

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