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• W4237985650 abstract "We investigate radial solutions for the problem <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout Enlarged left-brace 1st Row 1st Column minus normal upper Delta upper U equals StartFraction lamda plus delta StartAbsoluteValue nabla upper U EndAbsoluteValue squared Over 1 minus upper U EndFraction comma upper U greater-than 0 2nd Column a m p semicolon in upper B comma 2nd Row 1st Column upper U equals 0 2nd Column a m p semicolon on partial-differential upper B comma EndLayout> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=left left rowspacing=.2em columnspacing=1em displaystyle=false> <mml:mtr> <mml:mtd> <mml:mstyle displaystyle=true scriptlevel=0> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mi>U</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>+</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>∇<!-- ∇ --></mml:mi> <mml:mi>U</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>U</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mspace width=thickmathspace /> <mml:mi>U</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mstyle> </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:mtext> </mml:mtext> <mml:mi>B</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:mtext> </mml:mtext> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=true stretchy=true symmetric=true /> </mml:mrow> <mml:annotation encoding=application/x-tex>begin {cases} displaystyle -Delta U=frac {lambda +delta |nabla U|^2}{1-U},; U>0 & text {in} B, U=0 & text {on} partial B, end {cases}</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=upper B subset-of double-struck upper R Superscript upper N> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>Bsubset mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper N greater-than-or-equal-to 2 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(Ngeq 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the open unit ball and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda comma delta greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda , delta >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are real numbers. Two classes of solutions are considered in this work: (i) <italic>regular solutions</italic>, which satisfy <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than upper U greater-than 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>U</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0>U>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=application/x-tex>B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and (ii) <italic>rupture solutions</italic>, which satisfy <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper U left-parenthesis 0 right-parenthesis equals 1> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>U(0)=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and thus make the equation singular at the origin. Bifurcation with respect to parameter <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 also discussed." @default.
• W4237985650 created "2022-05-12" @default.
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• W4237985650 date "2022-01-20" @default.
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• W4237985650 title "Radial regular and rupture solutions for a PDE problem with gradient term and two parameters" @default.
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• W4237985650 doi "https://doi.org/10.1090/proc/15861" @default.
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