Global structure of positive solutions for superlinear $2m$th-boundary value problems

Ma, Ruyun ; An, Yulian

Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 161-172.

Résumé

We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem $$ \begin{aligned} (-1)^mu^{(2m)}(t)=\lambda a(t)f(u(t)),\ \ \ \ \ 01, \\ u^{(2i)}(0)=u^{(2i)}(1)=0,\ \ \ \ i=0,1,2,\cdots ,m-1 . \end{aligned} $$ where $a\in C([0,1], [0,\infty ))$ and $a(t_0)>0$ for some $t_0\in [0,1]$, $f\in C([0,\infty ),[0,\infty ))$ and $f(s)>0$ for $s>0$, and $f_0=\infty $, where $f_0=\lim _{s\rightarrow 0^+}f(s)/s$. We investigate the global structure of positive solutions by using Rabinowitz's global bifurcation theorem.

Détail

Classification : 34B08, 34B10, 34B18, 34G20, 47J15, 47N20
Keywords: multiplicity results; Lidstone boundary value problem; eigenvalues; bifurcation methods; positive solutions