Differential and Integral Equations

Some remarks about the Fucik spectrum and application to equations with jumping nonlinearities

Chakib Abchir

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Abstract

Let $L$ be a selfadjoint operator with compact resolvent and $\lambda $ an eigenvalue of $L$. When $\lambda $ is simple, it is well known that the Fucik spectrum $\Sigma $ near $\lambda $ consists of two nonincreasing curves. In this paper, we show that when $\lambda $ is not simple, $\Sigma $ contains two nonincreasing curves such that all points above or under both curves are not in $\Sigma $. After that, we give some existence results of solutions of the equation $Lu=\alpha u^{+}-\beta u^{-}+g(.,u)$ where $u^{\pm }=\max (0,\pm u)$.

Article information

Source
Differential Integral Equations, Volume 15, Number 9 (2002), 1045-1060.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060762

Mathematical Reviews number (MathSciNet)
MR1919761

Zentralblatt MATH identifier
1022.35032

Subjects
Primary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory
Secondary: 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05]

Citation

Abchir, Chakib. Some remarks about the Fucik spectrum and application to equations with jumping nonlinearities. Differential Integral Equations 15 (2002), no. 9, 1045--1060. https://projecteuclid.org/euclid.die/1356060762


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