## Differential and Integral Equations

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

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

#### 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