Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 100, pp. 1-8.
Title: Multiplicity results for fourth-order boundary-value problem at
resonance with variable coefficients
Author: Ling Xu (Northwest Normal Univ., Lanzhou, Gansu, China)
Abstract:
This paper studies the multiplicity of solutions for the fourth-order
boundary value problem at resonance with variable coefficients
$$\displaylines{
u^{(4)}+\beta(t)u''-\lambda_1u=g(t, u)+h(t),\quad t\in(0, 1),\cr
u(0)=u(1)=u''(0)=u''(1)=0,
}$$
where $\beta\in C[0,1]$ with $\beta(t)<\pi^2$ on $[0,1]$,
$g:[0, 1]\times \mathbb{R}\to \mathbb{R}$ is bounded
continuous function, $h\in L^2(0,1)$ and $\lambda_1>0$ is the
first eigenvalue of the associated linear homogeneous boundary value
problem
$$\displaylines{
u^{(4)}+\beta(t)u''-\lambda u=0,\quad t\in(0, 1),\cr
u(0)=u(1)=u''(0)=u''(1)=0.
}$$
The proof of our main result is based on the connectivity
properties of the solution sets of parameterized families of compact
vector fields.
Submitted April 14, 2008. Published July 30, 2008.
Math Subject Classifications: 39A10.
Key Words: Connected subsets; resonance; multiplicity results.