Symmetry and monotonicity of solutions for a system

Abstract In our thesis, we employ a forth and means of traveling flats for an equation set including completely non-linear non-local operators in B 1 (0). Under the conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ζ</mml:mi> <mml:mfenced> <mml:mi>η</mml:mi> </mml:mfenced> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>l</mml:mi> <mml:mi>o</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>υ</mml:mi> <mml:mfenced> <mml:mi>η</mml:mi> </mml:mfenced> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>l</mml:mi> <mml:mi>o</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> , the symmetric character of radial direction and monotone character of plus solutions for an equation set are proved. In order to get this result, we use the extreme value theory of narrow area for the equation set and pivotal ingredients for carrying on the way of traveling flats.