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

We study the structure of solution set of the nonlinear two-point boundary value problem{u″(x)+fλ(u(x))=0,−10λ>0 is a bifurcation parameter and fλ(u)=λ∑i=1maiuqi+∑j=1nbjupj satisfies (A1)–(A4). Under (A1)–(A4), we prove that there exists λ*>0λ*>0 such that the problem has exactly two positive solutions for 0<λ<λ*0<λ<λ*, exactly one positive solution for λ=λ*λ=λ*, and no positive solution for λ>λ*λ>λ*. More precisely, we give a complete description of the structure of the solution set.