Bifurcation diagrams of one-dimensional Kirchhoff-type equations
نویسندگان
چکیده
Abstract We study the one-dimensional Kirchhoff-type equation − ( b + a ‖ u accent="false">′ 2 ) accent="true">″ x = λ p , ∈ I ≔ − 1 > 0 ± -\left(b+a\Vert u^{\prime} {\Vert }^{2}){u}^{^{\prime\prime} }\left(x)=\lambda u{\left(x)}^{p},\hspace{1em}x\in I:= \left(-1,1),\hspace{1em}u\left(x)\gt 0,\hspace{1em}x\in I,\hspace{1em}u\left(\pm 1)=0, where xmlns:m="http://www.w3.org/1998/Math/MathML"> ∫ d width="0.1em" / \Vert ={\left({\int }_{I}u^{\prime} {\left(x)}^{2}{\rm{d}}x\right)}^{1\text{/}2} , a\gt 0,b\gt 0,p\gt 0 are given constants and \lambda \gt is a bifurcation parameter. establish exact solution {u}_{\lambda }\left(x) complete shape of curves ξ =\lambda \left(\xi ) ∞ \xi := }{\Vert }_{\infty } . also nonlinear eigenvalue problem μ -\Vert }^{p-1}{u}^{^{\prime\prime} }\left(x)=\mu u{\left(x)}^{p},x\in I,\hspace{1em}u\left(x)\gt 0,x\in p\gt 1 constant \mu an obtain first eigenfunction this explicitly by using simple time map method.
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ژورنال
عنوان ژورنال: Advances in Nonlinear Analysis
سال: 2022
ISSN: ['2191-950X', '2191-9496']
DOI: https://doi.org/10.1515/anona-2022-0265