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Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type

  • * Corresponding author: Yutian Lei

    * Corresponding author: Yutian Lei

This research is supported by the National Natural Science Foundation of China (11871278, 11671209)

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  • This paper is concerned with the existence/nonexistence of positive solutions of a weighted Hardy-Littlewood-Sobolev type integral system. Such a system is related to the extremal functions of the weighted Hardy-Littlewood-Sobolev inequality. The Serrin-type condition is critical for existence of positive solutions in $ L_{loc}^\infty(R^n \setminus \{0\}) $. When the Serrin-type condition does not hold, we prove the nonexistence by an iteration process. In addition, we find three pairs of radial solutions when the Serrin-type condition holds. One is singular, and the other two are integrable in $ R^n $ and decaying fast and slowly respectively.

    Mathematics Subject Classification: 45E10, 45G05, 45M20.


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