Advances in Differential Equations

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

A. M. Micheletti and C. Saccon

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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

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651634

Mathematical Reviews number (MathSciNet)
MR1919701

Zentralblatt MATH identifier
1034.58009

Subjects
Primary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)
Secondary: 35Q72 47J30: Variational methods [See also 58Exx]

Citation

Micheletti, A. M.; Saccon, C. Multiple solutions for an asymptotically linear problem with nonlinearity crossing a finite number of eigenvalues and application to a beam equation. Adv. Differential Equations 7 (2002), no. 10, 1193--1214. https://projecteuclid.org/euclid.ade/1356651634


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