Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

Abstract

Let λ * > 0 denote the largest possible value of λ such thathas a solution, where B is the unit ball in R n centered at the origin, p > n+4 n−4 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, in which case u * (x) ≤ r − 4 p−1 − 1 on the unit ball and actually solve part of the open problem which [9] left.