Differential and Integral Equations

On an open problem of Ambrosetti, Brezis and Cerami

Yuanji Cheng

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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

Subjects
Primary: 34B15: Nonlinear boundary value problems
Secondary: 34B18: Positive solutions of nonlinear boundary value problems 34C23: Bifurcation [See also 37Gxx] 35B32: Bifurcation [See also 37Gxx, 37K50] 35J60: Nonlinear elliptic equations 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05]

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


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