Bifurcation diagrams of one-dimensional Kirchhoff type equations

DOI10.1515/ANONA-2022-0265arXiv2109.14864MaRDI QIDQ6379000

Tetsutaro Shibata

Publication date: 30 September 2021

Abstract: We study the one-dimensional Kirchhoff type equation ${\displaystyle }$ -(b + aVert u'Vert^{2}) u(x) = lambda u(x)^p, x in I:= (-1,1), enskip u(x) > 0, enskip xin I, enskip u(pm 1) = 0, ${\displaystyle }$ where $Vert{u}^{\prime }Vert=left(in{t}_I{u}^{\prime }(x{)}^{2}dxight{)}^{1/2}$, $a>0,b>0,p>0$ are given constants and $lambda>0$ is a bifurcation parameter. We establish the exact solution $u_{l}ambda(x)$ and complete shape of the bifurcation curves $lambda=lambda(xi)$, where $xi:=Vert{u}_{l}ambdaVer{t}_{i}nfty$. We also study the nonlinear eigenvalue problem ${\displaystyle }$ -Vert u'Vert^{p-1} u(x) = mu u(x)^p, x in I, enskip u(x) > 0, xin I, enskip u(pm 1) = 0, ${\displaystyle }$ where $p>1$ is a given constant and $mu>0$ is an eigenvalue parameter. We establish the first eigenvalue and eigenfunction of this problem by using a simple time map method.

Mathematics Subject Classification ID

Bifurcation theory for ordinary differential equations (34C23) Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators (34L10) Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators (34L15) Boundary eigenvalue problems for ordinary differential equations (34B09)

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