Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters. (English) Zbl 1135.34007

Liu, Xi-Lan; Li, Wan-Tong

Math. Comput. Modelling 46, No. 3-4, 525-534 (2007).

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 \(f:[0,1]\times \mathbb R\to\mathbb R\) is continuous, \(\zeta,\eta\) and \(\lambda\in\mathbb R\) 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 \(\lambda\in \mathbb R\) 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 \( n_*\) \((n_*\in\mathbb N)\) distinct pairs of solutions, which depends on the fact that the pair \((\eta,\zeta)\) is located in the second or fourth (first) quadrant.

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
47J30	Variational methods involving nonlinear operators

Keywords:

existence; multiple solutions; fourth-order boundary value problem

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