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- W2912656087 abstract "The aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear problem. <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mo stretchy=false>{</mml:mo><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>1</mml:mn><mml:mo>-</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mfenced separators=|><mml:mrow><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=normal>′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi mathvariant=normal>′</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=false>(</mml:mo><mml:mn mathvariant=normal>0,1</mml:mn><mml:mo stretchy=false>)</mml:mo><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=normal>lim</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>→</mml:mo><mml:mn mathvariant=normal>0</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo></mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=normal>′</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn mathvariant=normal>0</mml:mn><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mn mathvariant=normal>1</mml:mn><mml:mo stretchy=false>)</mml:mo><mml:mo>=</mml:mo><mml:mn mathvariant=normal>0</mml:mn><mml:mo stretchy=false>}</mml:mo></mml:math>, where <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn mathvariant=normal>3</mml:mn><mml:mo>,</mml:mo></mml:math><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mi>σ</mml:mi><mml:mo><</mml:mo><mml:mn mathvariant=normal>1</mml:mn><mml:mo>,</mml:mo></mml:math> and <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:math> denotes a nonnegative continuous function that might have the property of being singular at <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant=normal>0</mml:mn></mml:math> and /or <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:math> and which satisfies certain condition associated to Karamata class. We emphasize that the nonlinearity might also be singular at <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant=normal>0</mml:mn></mml:math>, while the solution could blow-up at <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mrow><mml:mn mathvariant=normal>0</mml:mn></mml:mrow></mml:math>. Our method is based on the global estimates of potential functions and the Schauder fixed point theorem." @default.
- W2912656087 created "2019-02-21" @default.
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- W2912656087 date "2019-02-03" @default.
- W2912656087 modified "2023-10-16" @default.
- W2912656087 title "Existence and Global Asymptotic Behavior of Singular Positive Solutions for Radial Laplacian" @default.
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- W2912656087 doi "https://doi.org/10.1155/2019/3572132" @default.
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