Bifurcation diagrams of one-dimensional Kirchhoff-type equations

Abstract

We study the one-dimensional Kirchhoff-type equation −(b+a‖u′‖2)u″(x)=λu(x)p,x∈I≔(−1,1),u(x)>0,x∈I,u(±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 ‖u′‖=∫Iu′(x)2dx1/2\Vert u^{\prime} \Vert ={\left({\int }_{I}u^{\prime} {\left(x)}^{2}{\rm{d}}x\right)}^{1\text{/}2}, a>0,b>0,p>0a\gt 0,b\gt 0,p\gt 0 are given constants and λ>0\lambda \gt 0 is a bifurcation parameter. We establish the exact solution uλ(x){u}_{\lambda }\left(x) and complete shape of the bifurcation curves λ=λ(ξ)\lambda =\lambda \left(\xi ), where ξ≔‖uλ‖∞\xi := \Vert {u}_{\lambda }{\Vert }_{\infty }. We also study the nonlinear eigenvalue problem −‖u′‖p−1u″(x)=μu(x)p,x∈I,u(x)>0,x∈I,u(±1)=0,-\Vert u^{\prime} {\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 I,\hspace{1em}u\left(\pm 1)=0, where p>1p\gt 1 is a given constant and μ>0\mu \gt 0 is an eigenvalue parameter. We obtain the first eigenvalue and eigenfunction of this problem explicitly by using a simple time map method.

Keywords

• kirchhoff-type equations
• bifurcation curves
• exact value of the first eigenvalue
• 34c23
• 37g99