Inverse bifurcation problems for nonlinear Sturm–Liouville problems

Abstract

We consider the inverse nonlinear eigenvalue problems for the equation which is motivated biologically by the problem of population dynamics. It is assumed that f(u) is an unknown nonlinear term. Under the standard growth conditions on f, for any given α > 0, there exists a unique solution of the equation with ‖uα‖q = α, where ‖ ⋅ ‖q denotes usual Lq-norm. This curve λ(q, α) is called the Lq-bifurcation curve. We show that (i) the unknown nonlinear term is determined as f(u) = up (p > 1) nearly exponentially for u ≫ 1 under the reasonable condition on λ(q, α) for α ≫ 1, and (ii) by variational method, the unknown nonlinear term is determined uniquely for u ⩾ 0 under the additional conditions on f when q = 2.