Super Polyharmonic Property of Solutions for PDE Systems and Its Applications

By Wenxiong CHEN *, Congming LI #. 2011

73 citations recorded by the source corpus.

In this paper, we prove that all the positive solutions for the PDE system (-\Delta)^{k}u_{i} = f_{i}(u_{1},..., u_{m}), x \in R^{n}, i = 1, 2,..., m are super polyharmonic, i.e. (-\Delta)^{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-\alpha}}f_{i}(u_{1}(y),..., u_{m}(y))dy, x \in R^{n} and PDE system when \alpha? = 2k < n

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6a2075a699e5a70076a0f06f28f3030bfc20afc7

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In Applied Mathematics

1. Yutian Lei
2. Congming LI #
3. Wenxiong CHEN *
4. Li Ma
5. Fengbo Hang
6. Dezhong Chen
7. Xiaohui Yu
8. Chao Ma

In Geometry and Topology

1. Yutian Lei
2. Congming LI #
3. Wenxiong CHEN *
4. Li Ma
5. Tianling Jin
6. Jingang Xiong
7. Wolfgang Reichel

Citations in the corpus

• Sharp criteria of Liouville type for some nonlinear systems (2013, background): “The nonexistence of the subcritical case is a corollary of the Liouville theorem in [54] and the equivalence result between PDEs and IEs in [9].”