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Boundary value problems for some fourth order ordinary differential equations. (English) Zbl 0682.34020
The following nonlinear beam equation is studied (1) $$u^{(4)}-\alpha (x)u+f(x,u)=h(x),$$ $$x\in (0,\pi)$$ under one of the boundary conditions (2) $$u(0)=u(\pi)=u''(0)=u''(\pi)=0$$, or (3) $$u(0)=u(\pi)=u'(0)=u'(\pi)=0$$. Here $$\alpha$$ (x) is a bounded, nonnegative function and it is assumed that 0 is the first eigenvalue of the linear part of (1). The right-hand side h(x) is orthogonal to the first eigenfunction. Using variational methods, some existence and multiplicity results are obtained: (a) existence of at least one solution if f(x,u) satisfies an assumption which generalizes a “sign condition”; (b) existence of at least one solution if f(x,u) satisfies a Landesman- Lazer condition: (c) existence of at least three solutions near resonance (a variational approach to bifurcation from infinity), for the problem (1)-(2).
Reviewer: L.Sanchez

##### MSC:
 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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