Properties of the extremal solution for a fourth-order elliptic problem

2010/09/14 by Lai, Baishun, Du, Zhuoran
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.2546

Abstract

Let λ^*>0 denote the largest possible value of λ such that \arraylllllll Δ²u=\fracλ(1-u)^p & \in B, 01 and n is the exterior unit normal vector. We show that for λ=λ^* this problem possesses a unique weak solution u^*, called the extremal solution. We prove that u^* is singular when n≥ 13 for p large enough and 1-C₀r^(4)/(p+1)≤ u^*(x)≤ 1-r^(4)/(p+1) on the unit ball, where C₀:=(λ^*/λ)^(1)/(p+1) and λ:=\frac8(p-1)(p+1)²[n-(2(p-1))/(p+1)][n-(4p)/(p+1)]. Our results actually complete part of the open problem which \citeD lef

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