On an open problem of Ambrosetti, Brezis and Cerami

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).