NONEXISTENCE AND SYMMETRY OF SOLUTIONS FOR SCHR ?DINGER SYSTEMS INVOLVING FRACTIONAL LAPLACIAN

Ran Zhuo; Yan Li

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NONEXISTENCE AND SYMMETRY OF SOLUTIONS FOR SCHR ?DINGER SYSTEMS INVOLVING FRACTIONAL LAPLACIAN

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In this paper, we consider the following Schr?dinger systems involving pseudo-differential operator in R~n where α and γ are any number between 0 and 2, α does not identically equal to γ. We employ a direct method of moving planes to partial differential equations (PDEs) (1). Instead of using the Caffarelli-Silvestre's extension method and the method of moving planes in integral forms, we directly apply the method of moving planes to the nonlocal fractional order pseudo-differential system. We obtained radial symmetry in the critical case and non-existence in the subcritical case for positive solutions. In the proof, combining a new approach and the integral definition of the fractional Laplacian, we derive the key tools, which are needed in the method of moving planes, such as, narrow region principle, decay at infinity. The new idea may hopefully be applied to many other problems.

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《Discrete and continuous dynamical systems》 |2019年第3期|1595-1611|共17页
作者
Ran Zhuo; Yan Li;
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作者单位

School of mathematics and statistics Huanghuai University Zhumadian, Henan 463000, China;

Department of Mathematics Baylor University Waco, TX 76798, USA;

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正文语种英语
中图分类理论力学（一般力学）;
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Schr?dinger systems; fractional Laplacian; narrow region principle; decay at infinity; method of moving planes; Kelvin transform; radial symmetry; nonexistence of positive solutions;

NONEXISTENCE AND SYMMETRY OF SOLUTIONS FOR SCHR ?DINGER SYSTEMS INVOLVING FRACTIONAL LAPLACIAN

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