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Super polyharmonic property of solutions for PDE systems and its applications

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


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