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- W3196796896 abstract "Abstract We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mfenced open={ close=> <m:mrow> <m:mtable displaystyle=true> <m:mtr> <m:mtd columnalign=left> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mfenced open=( close=)> <m:mrow> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo accent=true>′</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:msqrt> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo accent=true>′</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msqrt> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mo accent=true>′</m:mo> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:msup> <m:mrow> <m:mfenced open=( close=)> <m:mrow> <m:mfrac> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width=1.0em /> <m:mo>−</m:mo> <m:mi>L</m:mi> <m:mo><</m:mo> <m:mi>x</m:mi> <m:mo><</m:mo> <m:mi>L</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=left> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi>L</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> left{begin{array}{l}-{left(frac{{u}^{^{prime} }}{sqrt{1+{u}^{^{prime} 2}}}right)}^{^{prime} }=lambda {left(frac{u}{1+u}right)}^{p},hspace{1.0em}-Llt xlt L, uleft(-L)=uleft(L)=0,end{array}right. where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>λ</m:mi> </m:math> lambda is a bifurcation parameter, and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>L</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> L,pgt 0 are two evolution parameters. We prove that on the <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>,</m:mo> <m:mo>‖</m:mo> <m:mi>u</m:mi> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> left(lambda ,Vert u{Vert }_{infty }) -plane, for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mfrac> <m:mrow> <m:msqrt> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msqrt> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:mfrac> </m:math> 0lt ple frac{sqrt{2}}{4} , the bifurcation curve is <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>⊃</m:mo> </m:math> supset -shaped bifurcation starting from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> left(0,0) . And for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mfrac> </m:math> p=1,fleft(u)=frac{u}{1+u} is a logistic function, then the bifurcation curve is also <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>⊃</m:mo> </m:math> supset -shaped bifurcation starting from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mfenced open=( close=)> <m:mrow> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mn>4</m:mn> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mfenced> </m:math> left(frac{{pi }^{2}}{4{L}^{2}},0right) . While for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:math> pgt 1 , the bifurcation curve is reversed <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>ε</m:mi> </m:math> varepsilon -like shaped bifurcation if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>L</m:mi> <m:mo>></m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:math> Lgt {L}^{ast } , and is exactly decreasing for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>λ</m:mi> <m:mo>></m:mo> <m:msup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:math> lambda gt {lambda }^{ast } if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>L</m:mi> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msub> </m:math> 0lt Llt {L}_{ast } ." @default.
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- W3196796896 date "2021-01-01" @default.
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- W3196796896 title "On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type" @default.
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