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Zbl 1135.34007
Liu, Xi-Lan; Li, Wan-Tong
Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters.
(English)
[J] Math. Comput. Modelling 46, No. 3-4, 525-534 (2007). ISSN 0895-7177

Summary: This paper is concerned with the existence and multiplicity of the solutions for the fourth-order boundary value problem $$u^{(4)}(t)+\eta u''(t)-\zeta u(t)=\lambda f(t,u(t)),\quad 0<t<1,$$ $$u(0)=u(1)=u''(0)=u''(1)=0,$$ where $f:[0,1]\times \Bbb R\to\Bbb R$ is continuous, $\zeta,\eta$ and $\lambda\in\Bbb R$ are parameters. Using the variational structure of the above boundary value problem and critical point theory, it is shown that the different locations of the pair $(\eta,\zeta)$ and $\lambda\in \Bbb R$ lead to different existence results for the above boundary value problem. More precisely, if the pair $(\eta,\zeta)$ is on the left side of the first eigenvalue line, then the above boundary value problem has only the trivial solution for $\lambda\in (-\lambda,0)$ and has infinitely many solutions for $\lambda\in (0,\infty)$; if $(\eta,\zeta)$ is on the right side of the first eigenvalue line and $\lambda\in (-\infty,0)$, then the above boundary value problem has two nontrivial solutions or has at least $n_*$ $(n_*\in\Bbb N)$ distinct pairs of solutions, which depends on the fact that the pair $(\eta,\zeta)$ is located in the second or fourth (first) quadrant.
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE
47J30 Variational methods

Keywords: existence; multiple solutions; fourth-order boundary value problem

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