Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 27, pp. 1-7.

Title: Exact multiplicity of solutions for a class of
       two-point boundary value problems
       
Authors: Yulian An (Shanghai Institute of Technology, China)
         Ruyun Ma (Northwest Normal Univ., Lanzhou, China)

Abstract:
 We consider the exact multiplicity of nodal solutions of the
 boundary  value problem
 $$\displaylines{
 u''+\lambda f(u)=0 , \quad t\in (0, 1),\cr
 u'(0)=0,\quad u(1)=0,
 }$$
 where $\lambda \in \mathbb{R}$ is a positive parameter.
 $f\in C^1(\mathbb{R}, \mathbb{R})$ satisfies
 $f'(u)>\frac{f(u)}{u}$, if $u\neq 0$.
 There exist $\theta_1<s_1<0<s_2<\theta_2$ such that
 $f(s_1)=f(0)=f(s_2)=0$; $uf(u)>0$, if $u<s_1$ or $u>s_2$; $uf(u)<0$,
 if $s_1<u<s_2$ and $u\neq 0$; $\int_{\theta_1}^0
 f(u)du=\int_0^{\theta_2} f(u)du=0$. The limit
 $f_\infty=\lim_{s\to \infty} \frac{f(s)}{s}\in (0,\infty)$.
 Using bifurcation techniques and the Sturm comparison theorem,
 we obtain curves of solutions which bifurcate from infinity at the
 eigenvalues of the corresponding linear problem, and obtain the
 exact multiplicity of solutions to the problem for $\lambda$ lying
 in some interval in $\mathbb{R}$.

Submitted September 30, 2009. Published February 16, 2010.
Math Subject Classifications: 34B15, 34A23.
Key Words: Exact multiplicity; nodal solutions; bifurcation from infinity; 
           linear eigenvalue problem.