Bifurcation diagrams of one-dimensional Kirchhoff type equations

Abstract

We study the one-dimensional Kirchhoff type equation $$-(b+a\Vert u^{\prime }{\Vert }^2)u^{\prime \prime }(x)=\lambda u(x{)}^p,x\in I:=(-1,1),u(x)>0,x\in I,u(\pm 1)=0,$$ where $\Vert u^{\prime }\Vert ={\left({\int }_I{u}^{\prime }(x{)}^{2}dx\right)}^{1/2}$, $a>0,b>0,p>0$ are given constants and $\lambda >0$ is a bifurcation parameter. We establish the exact solution $u_{\lambda }(x)$ and complete shape of the bifurcation curves $\lambda =\lambda (\xi )$, where $\xi :=\Vert u_{\lambda }{\Vert }_{\infty }$. We also study the nonlinear eigenvalue problem $$-\Vert u^{\prime }{\Vert }^{p-1}{u}^{\prime \prime }(x)=\mu u(x{)}^p,x\in I,u(x)>0,x\in I,u(\pm 1)=0,$$ 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.