Differential and Integral Equations

On an open problem of Ambrosetti, Brezis and Cerami

Yuanji Cheng

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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

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

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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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