This paper investigates the global bifurcation diagrams and multiplicity of positive solutions for a one-dimensional prescribed mean curvature equation with cubic nonlinearity <Equation ID="Equ39"> <EquationSource Format="TEX">${\left\{ \begin{array}{ll} \!-\!\left( \dfrac{u'(x)}{\sqrt{1+(u'(x))^2}} \right) ' \!=\! \lambda (\!-\!\epsilon u^3+u^2+u+1), \quad \!-\!L \!<\! x \!<\! L,\\ u(-L) \!=\! u(L) \!=\! 0, \end{array}\right. }$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mspace width="-0.166667em" /> <mo>-</mo> <mspace width="-0.166667em" /> <msup> <mfenced close=")" open="("> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mstyle> </mfenced> <mo>′</mo> </msup> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mspace width="-0.166667em" /> <mo>-</mo> <mspace width="-0.166667em" /> <mi>ϵ</mi> <msup> <mi>u</mi> <mn>3</mn> </msup> <mo>+</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>u</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="-0.166667em" /> <mo>-</mo> <mspace width="-0.166667em" /> <mi>L</mi> <mspace width="-0.166667em" /> <mo><</mo> <mspace width="-0.166667em" /> <mi>x</mi> <mspace width="-0.166667em" /> <mo><</mo> <mspace width="-0.166667em" /> <mi>L</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mi>u</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda , L,\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <mi>ϵ</mi> </mrow> </math></EquationSource> </InlineEquation> are positive parameters. We prove that, for any evolution parameters <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\epsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the global bifurcation curve of positive solutions on the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\lambda , \Vert u\Vert _{\infty })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mo stretchy="false">‖</mo> <mi>u</mi> <msub> <mo stretchy="false">‖</mo> <mi>∞</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-plane is either strictly increasing, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\supset $</EquationSource> <EquationSource Format="MATHML"><math> <mo>⊃</mo> </math></EquationSource> </InlineEquation>-shaped, S-shaped, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\supset $</EquationSource> <EquationSource Format="MATHML"><math> <mo>⊃</mo> </math></EquationSource> </InlineEquation>-like shaped, or S-like shaped, depending on the parameter regimes.

Global Bifurcation and Exact Multiplicity for a 1D Cubic Mean Curvature Equation

期刊论文
发表日期：

2026年7月27日

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Hao Chen,
Rui Yang

摘要

This paper investigates the global bifurcation diagrams and multiplicity of positive solutions for a one-dimensional prescribed mean curvature equation with cubic nonlinearity ${\left\{ \begin{array}{ll} \!-\!\left( \dfrac{u'(x)}{\sqrt{1+(u'(x))^2}} \right) ' \!=\! \lambda (\!-\!\epsilon u^3+u^2+u+1), \quad \!-\!L \!<\! x \!<\! L,\\ u(-L) \!=\! u(L) \!=\! 0, \end{array}\right. }$ $\{\begin{matrix} - {(\frac{u^{'} (x)}{\sqrt{1 + {(u^{'} (x))}^{2}}})}^{'} = λ (- ϵ u^{3} + u^{2} + u + 1), - L < x < L, \\ u (- L) = u (L) = 0, \end{matrix})$ where $\lambda , L,\epsilon $ $λ, L, ϵ$ are positive parameters. We prove that, for any evolution parameters $\epsilon >0$ $ϵ > 0$ and $L > 0$ $L > 0$ , the global bifurcation curve of positive solutions on the $(\lambda , \Vert u\Vert _{\infty })$ $(λ, ‖ u ‖_{\infty})$ -plane is either strictly increasing, $\supset $ $\supset$ -shaped, S-shaped, $\supset $ $\supset$ -like shaped, or S-like shaped, depending on the parameter regimes.