## Differential and Integral Equations

### On an open problem of Ambrosetti, Brezis and Cerami

Yuanji Cheng

#### Abstract

In this paper we study the structure of all solutions to the boundary value problem, which is the open problem D of Ambrosetti, Brezis and Cerami [1] $$-u'' = \lambda |u|^{q-1}u+|u|^{p-1}u, \quad t\in [a, b], \ \ u(a)=u(b)=0,$$ where $0 <q <1 <p,$ $\lambda >0 .$ We obtain a complete characterization of its solutions and the bifurcation graph. By perturbation, we show also instablity of the structure of the solutions for the above problem (see Figure 3).

#### Article information

Source
Differential Integral Equations, Volume 15, Number 9 (2002), 1025-1044.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060761

Mathematical Reviews number (MathSciNet)
MR1919760

Zentralblatt MATH identifier
1029.34017

#### Citation

Cheng, Yuanji. On an open problem of Ambrosetti, Brezis and Cerami. Differential Integral Equations 15 (2002), no. 9, 1025--1044. https://projecteuclid.org/euclid.die/1356060761