### Multiple solutions for an asymptotically linear problem with nonlinearity crossing a finite number of eigenvalues and application to a beam equation

#### Abstract

We consider a semilinear problem of the type $Lu=f(b,u),$ where $f(b,u)\simeq bu$ as $u\to 0$ and $f(b,u)\simeq b_\infty u$ as $\|u \| \to\infty$ assuming that there exist a finite number of eigenvalues of the linear operator $L$ between $b$ and $b_\infty$. Under suitable assumptions we prove the existence of four nontrivial solutions for $b$ close to an eigenvalue. We give an application to problems of oscillations of a forced beam.

#### Article information

Source
Adv. Differential Equations, Volume 7, Number 10 (2002), 1193-1214.

Dates
First available in Project Euclid: 27 December 2012