We apply bifurcation theories to construct repeated S-shaped and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1201_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>-shaped global unbounded continua of positive solutions of the following kth mean curvature problems in Minkowski space <Equation ID="Equ43"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1201_Article_Equ43.gif" Format="GIF" Height="87" Rendition="HTML" Resolution="72" Type="Linedraw" Width="417" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} & -\Big (r^{N-k}\Big (\frac{u'}{\sqrt{1-u'^2}}\Big )^k\Big )'\\ & \quad =\lambda \frac{N}{C_N^k}r^{N-1}H_k(r,\ u),\quad r\in (0,\ R),\quad u(0)=0=u'(R), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <mo>-</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <msup> <mi>r</mi> <mrow> <mi>N</mi> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mfrac> <msup> <mi>u</mi> <mo>′</mo> </msup> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>u</mi> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msup> </mrow> </msqrt> </mfrac> <msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi>k</mi> </msup> <msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>′</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mspace width="1em" /> <mo>=</mo> <mi>λ</mi> <mfrac> <mi>N</mi> <msubsup> <mi>C</mi> <mi>N</mi> <mi>k</mi> </msubsup> </mfrac> <msup> <mi>r</mi> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>H</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mspace width="4pt" /> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>r</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="4pt" /> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1201_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">$3\le k<N<2k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>≤</mo> <mi>k</mi> <mo><</mo> <mi>N</mi> <mo><</mo> <mn>2</mn> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> with k, N being integers and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1201_Article_IEq3.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">$C_N^k=\frac{N!}{(N-k)!k!}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>C</mi> <mi>N</mi> <mi>k</mi> </msubsup> <mo>=</mo> <mfrac> <mrow> <mi>N</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> the combinational constant. Moreover, Calabi–Bernstein type asymptotic of one-sign solutions are given.

Global structure of multiple positive solutions for the kth mean curvature in Minkowski space

期刊论文
发表日期：

2025年5月12日

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Tianlan Chen,
Christopher S. Goodrich

摘要

We apply bifurcation theories to construct repeated S-shaped and $\Sigma $ $Σ$ -shaped global unbounded continua of positive solutions of the following kth mean curvature problems in Minkowski space $\begin{aligned} & -\Big (r^{N-k}\Big (\frac{u'}{\sqrt{1-u'^2}}\Big )^k\Big )'\\ & \quad =\lambda \frac{N}{C_N^k}r^{N-1}H_k(r,\ u),\quad r\in (0,\ R),\quad u(0)=0=u'(R), \end{aligned}$ $\begin{matrix} - (r^{N - k} (\frac{u^{'}}{\sqrt{1 - u^{' 2}}})^{k})^{'} \\ = λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k} (r, u), r \in (0, R), u (0) = 0 = u^{'} (R), \end{matrix}$ where $3\le k<N<2k$ $3 \leq k < N < 2 k$ with k, N being integers and $C_N^k=\frac{N!}{(N-k)!k!}$ $C_{N}^{k} = \frac{N!}{(N - k)! k!}$ the combinational constant. Moreover, Calabi–Bernstein type asymptotic of one-sign solutions are given.