Super Polyharmonic Property of Solutions for PDE Systems and Its Applications
arXiv:1110.2539
Abstract
In this paper, we prove that all the positive solutions for the PDE system (-Δ)^{k}u_{i} = f_{i}(u_{1},..., u_{m}), x \in R^{n}, i = 1, 2,..., m are super polyharmonic, i.e. (-Δ)^{j}u_{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) = \int_{R^{n}} \frac{1}{|x - y|^{n-α}}f_{i}(u_{1}(y),..., u_{m}(y))dy, x \in R^{n} and PDE system when α? = 2k < n