A characterization related to a two-point boundary value problem

Original · EN

In this short note, we establish the following result: Let f:[0,+∞[→ [0,+∞[, α:[0,1]→]0,+∞[be two continuous functions, with f(0)=0. Assume that, for some a>0, the function ξ→ ∫₀ξf(t)dt ξ² is non-increasing in]0,a]. Then, the following assertions are equivalent: (i) for each b>0, the function ξ→ ∫₀ξf(t)dt ξ² is not constant in]0,b]; (ii) for each r>0, there exists an open interval I]0,+∞[such that, for every λ∈ I, the problem -u''=λα(t)f(u) & in [0,1] & u>0 & in]0,1[& u(0)=u(1)=0 has a solution u satisfying ∫₀¹|u'(t)|²dt<r.