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• W2049067405 abstract "We will study the entire positive <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> solution of the geometrically and analytically interesting integral equation: <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mi>u</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mrow><mml:mi mathvariant=normal>‍</mml:mi></mml:mrow><mml:mo stretchy=false>|</mml:mo><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=false>|</mml:mo><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mi>d</mml:mi><mml:mi>y</mml:mi></mml:math> with <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>q</mml:mi></mml:math> in <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mrow><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>. We will show that only when <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>11</mml:mn></mml:math>, there are positive entire solutions which are given by the closed form <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mi>u</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo stretchy=false>|</mml:mo><mml:mi>x</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> up to dilation and translation. The paper consists of two parts. The first part is devoted to showing that <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math> must be equal to 11 if there exists a positive entire solution to the integral equation. The tool to reach this conclusion is the well-known Pohozev identity. The amazing cancelation occurred in Pohozev’s identity helps us to conclude the claim. It is this exponent which makes the moving sphere method work. In the second part, as normal, we adopt the moving sphere method based on the integral form to solve the integral equation." @default.
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• W2049067405 date "2013-07-24" @default.
• W2049067405 modified "2023-09-24" @default.
• W2049067405 title "Entire Solutions of an Integral Equation in R5" @default.
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• W2049067405 doi "https://doi.org/10.1155/2013/384394" @default.
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