The structure of the solution set of a generalized Ambrosetti–Brezis–Cerami problem in one space variable

Abstract

We study the structure of solution set of the nonlinear two-point boundary value problem { u ″ ( x ) + f λ ( u ( x ) ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ > 0 is a bifurcation parameter and f λ ( u ) = λ ∑ i = 1 m a i u q i + ∑ j = 1 n b j u p j satisfies (A1)–(A4). Under (A1)–(A4), we prove that there exists λ * > 0 such that the problem has exactly two positive solutions for 0 < λ < λ * , exactly one positive solution for λ = λ * , and no positive solution for λ > λ * . More precisely, we give a complete description of the structure of the solution set.