Qualitative properties of solutions for an integral equation

By Wenxiong CHEN *, Congming LI #, B. Ou. 2003

214 citations recorded by the source corpus.

Let $n$ be a positive integer and let $ 0 < \alpha < n.$ In this paper, we study more general integral equation $ u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} K(y) u(y)^p dy. We establish regularity, radial symmetry, and monotonicity of the solutions. We also consider subcritical cases, super critical cases, and singular solutions in all cases; and obtain qualitative properties for these solutions.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e698f09c2733245d7dffd012e74f8f0cca84fe0e

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In Applied Mathematics

1. Wenxiong CHEN *
2. Congming LI #
3. Li Ma
4. Yutian Lei
5. Dezhong Chen
6. G. Lu
7. Patricio Felmer
8. Y. Wang

In Geometry and Topology

1. Congming LI #
2. Wenxiong CHEN *
3. Li Ma
4. Yutian Lei
5. Patricio Felmer
6. G. Lu
7. Dezhong Chen
8. Y. Wang

Citations in the corpus

• Radial symmetry of positive solutions to equations involving the fractional laplacian (2013, methodology): “Great attention has also been devoted to symmetry results for equations involving the fractional Laplacian in R , such as in the work by Li [19] and Chen, Li and Ou [8, 9], where the method of moving planes in integral form has been developed to treat various equations and systems, see also Ma and C”
• The best constant in a weighted Hardy-Littlewood-Sobolev inequality (2007, methodology): “To establish the symmetry of the solution, Chen, Li, and Ou ([4], [5], [6]) introduced a new idea, an integral form of the method of moving planes.”
• CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS (2009, methodology): “To establish the symmetry of the solutions to (7), Chen, Li, and Ou [6]–[8] introduced a new idea, an integral form of the method of moving planes.”
• A Liouville type Theorem for an integral system (2006, methodology): “We shall use Chen-Li-Ou’s method [3] [4] to prove this theorem.”
• Super Polyharmonic Property of Solutions for PDE Systems and Its Applications (2011, methodology): “There are many more applications of this method to integral systems, and the interested readers may see [10] [12] [15] [20] [21] [28] [33] [34] [40] [42] [45] [46] [47] [50] [53] [54] and the references therein.”
• Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy–Littlewood–Sobolev system of integral equations (2012, background): “9 and the corresponding partial differential equations have been studied extensively in the last 40 years, see more details in [2,3,5–11,13–22].”