Electron. J. Diff. Eqns., Vol. 2008(2008), No. 100, pp. 1-8.

Multiplicity results for fourth-order boundary-value problem at resonance with variable coefficients

Ling Xu

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.

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Ling Xu
College of Mathematics and Information Science
Northwest Normal University
Lanzhou, Gansu 730070, China
email: xuling_216@yahoo.cn

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