Exact Multiplicity of Solutions for a Class of Two-point Boundary Value Problems

We consider the exact multiplicity of nodal solutions of the boundary value problem u′′ + λf(u) = 0, t ∈ (0, 1), u′(0) = 0, u(1) = 0, where λ ∈ R is a positive parameter. f ∈ C1(R,R) satisfies f ′(u) > f(u) u , if u 6= 0. There exist θ1 < s1 < 0 < s2 < θ2 such that f(s1) = f(0) = f(s2) = 0; uf(u) > 0, if u < s1 or u > s2; uf(u) < 0, if s1 < u < s2 and u 6= 0; R 0 θ1 f(u)du = R θ2 0 f(u)du = 0. The limit f∞ = lims→∞ f(s) s ∈ (0,∞). Using bifurcation techniques and the Sturm comparison theorem, we obtain curves of solutions which bifurcate from infinity at the eigenvalues of the corresponding linear problem, and obtain the exact multiplicity of solutions to the problem for λ lying in some interval in R.