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• W4210252465 abstract "We consider the following Gelfand problem <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper P right-parenthesis Subscript lamda Baseline StartLayout Enlarged left-brace 1st Row 1st Column minus normal upper Delta u equals lamda a left-parenthesis x right-parenthesis f left-parenthesis u right-parenthesis 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u greater-than 0 2nd Column a m p semicolon in normal upper Omega comma 3rd Row 1st Column u equals 0 2nd Column a m p semicolon on partial-differential normal upper Omega comma EndLayout> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>P</mml:mi> <mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> </mml:msub> <mml:mspace width=2em /> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=left left rowspacing=4pt columnspacing=1em> <mml:mtr> <mml:mtd> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>a</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> in </mml:mtext> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> in </mml:mtext> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext> on </mml:mtext> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=true stretchy=true symmetric=true /> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} (P)_lambda qquad left {begin {array}{ll} -Delta u = lambda a(x) f(u) & text { in } Omega , u>0 & text { in } Omega , u= 0 & text { on } partial Omega , end{array}right . end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a parameter and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis u right-parenthesis equals e Superscript u> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>u</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>f(u)=e^u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis u right-parenthesis equals left-parenthesis u plus 1 right-parenthesis Superscript p> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>f(u)=(u+1)^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>a(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a nonnegative function with certain monotonicity (we allow <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a left-parenthesis x right-parenthesis equals 1> <mml:semantics> <mml:mrow> <mml:mi>a</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:mrow> <mml:annotation encoding=application/x-tex>a(x)=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Here <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega> <mml:semantics> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:annotation encoding=application/x-tex>Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an annular domain which is also a double domain of revolution. Our interest will be in the question of the regularity of the extremal solution. We obtain improved compactness because of the annular nature of the domain and we obtain further compactness under some monotonicity assumptions on the domain." @default.
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• W4210252465 date "2022-05-13" @default.
• W4210252465 modified "2023-09-30" @default.
• W4210252465 title "The Gelfand problem on annular domains of double revolution with monotonicity" @default.
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• W4210252465 doi "https://doi.org/10.1090/proc/15990" @default.
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