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

An, Yulian ; Ma, Ruyun

Electronic journal of differential equations, Tome 2010 (2010)

Résumé

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 $\theta_1$ such that $f(s_1)=f(0)=f(s_2)=0; uf(u)>0$, if $u$ or $u>s_2; uf(u)0$, if $s_1$ 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 $\mathbb{R}$.

Détail

Classification : 34B15, 34A23
Keywords: exact multiplicity, nodal solutions, bifurcation from infinity, linear eigenvalue problem