Bifurcation diagrams of one-dimensional Kirchhoff-type equations
Tetsutaro Shibata · Advances in Nonlinear Analysis · 2022
Abstract We study the one-dimensional Kirchhoff-type equation − ( b + a ‖ u ′ ‖ 2 ) u ″ ( x ) = λ u ( 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 ′ ‖ = ∫ I u ′ ( x ) 2 d x 1 / 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 > 0 a\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