Super Polyharmonic Property of Solutions for PDE Systems and Its Applications

2011/10/12 by Chen, Wenxiong, Li, Congming
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1110.2539

Abstract

In this paper, we prove that all the positive solutions for the PDE system (-Δ)^ku_i = f_i(u₁,..., u_m), x ∈ Rⁿ, i = 1, 2,..., m are super polyharmonic, i.e. (-Δ)^ju_i > 0, j = 1, 2,..., k - 1; i = 1, 2,...,m. To prove this important super polyharmonic property, we introduced a few new ideas and derived some new estimates. As an interesting application, we establish the equivalence between the integral system u_i(x) = ∫_Rⁿ \frac1|x - y|^n-αf_i(u₁(y),..., u_m(y))dy, x ∈ Rⁿ and PDE system when α? = 2k < n

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