NONEXISTENCE AND SYMMETRY OF SOLUTIONS TO SOME FRACTIONAL LAPLACIAN EQUATIONS IN THE UPPER HALF SPACE

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doi:10.1016/S0252-9602(17)30040-1

Abstract:

In this article, we consider the fractional Laplacian equation
where 0 < α < 2, R₊ⁿ:={x=(x₁, x₂,…, x_n)|x_n > 0}. When K is strictly decreasing with respect to|x'|, the symmetry of positive solutions is proved, where x'=(x₁, x₂, …, x_n-1) ∈ R^n-1. When K is strictly increasing with respect to x_n or only depend on x_n, the nonexistence of positive solutions is obtained.

Key words: Fractional Laplacian, method of moving planes, radial symmetry, nonexistence

Yanyan GUO. NONEXISTENCE AND SYMMETRY OF SOLUTIONS TO SOME FRACTIONAL LAPLACIAN EQUATIONS IN THE UPPER HALF SPACE[J].Acta mathematica scientia,Series B, 2017, 37(3): 836-851.

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