Electron. J. Diff. Equ., Vol. 2010(2010), No. 27, pp. 1-7.

Exact multiplicity of solutions for a class of two-point boundary value problems

Yulian An, Ruyun Ma

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.

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Yulian An
Department of Mathematics
Shanghai Institute of Technology
shanghai 200235, China
email: an_yulian@tom.com
Ruyun Ma
Department of Mathematics
Northwest Normal University
Lanzhou 730070, Gansu, China
email: mary@nwnu.edu.cn

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