Super polyharmonic property of solutions for PDE systems and its applications

Wenxiong Chen; Congming Li

doi:10.3934/cpaa.2013.12.2497

Article Contents

2013, Volume 12, Issue 6: 2497-2514. Doi: 10.3934/cpaa.2013.12.2497

This issue Previous Article Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations Next Article Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations

Super polyharmonic property of solutions for PDE systems and its applications

Wenxiong Chen^1, and
Congming Li^2,

1.
College of Mathematics and Information Science, Henan Normal University
2.
Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240

Received: May 2012

Revised: December 2012

Published: November 2013

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we prove that all the positive solutions for the PDE system \begin{eqnarray} (- \Delta)^k u_i = f_i(u_1, \cdots, u_m), \ x \in R^n, \ i = 1, 2, \cdots, m \ \ \ \ \ (1) \end{eqnarray} are super polyharmonic, i.e. \begin{eqnarray} (- \Delta)^j u_i > 0, \ j=1, 2, \cdots, k-1; \ i =1, 2, \cdots, m. \end{eqnarray}
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 \begin{eqnarray} u_i(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} f_i(u_1(y), \cdots, u_m(y)) d y, \ x \in R^n \ \ \ \ \ (2) \end{eqnarray} and PDE system (1) when $\alpha = 2k < n.$

In the last few years, a series of results on qualitative properties for solutions of integral systems (2) have been obtained, since the introduction of a powerful tool--the method of moving planes in integral forms. Now due to the equivalence established here, all these properties can be applied to the corresponding PDE systems.

We say that systems (1) and (2) are equivalent, if whenever $u$ is a positive solution of (2), then $u$ is also a solution of \begin{eqnarray} (- \Delta)^k u_i = c f_i(u_1, \cdots, u_m), \ x \in R^n, \ i= 1,2, \cdots, m \end{eqnarray} with some constant $c$; and vice versa.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 45G15.

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