Inverse bifurcation problems for nonlinear Sturm-Liouville problems

We consider the inverse nonlinear eigenvalue problems for the equation -u ''(t) + f(u(t)) = lambda u(t), u(t) > 0, t is an element of I =: (0, 1), u(0) = u(1) = 0 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 alpha > 0, there exists a unique solution (lambda, u) = (lambda(q, alpha), u(alpha)) is an element of R+ x C-2(I) over bar of the equation with parallel to u(alpha)parallel to(q) = alpha, where parallel to.parallel to(q) denotes usual L-q-norm. This curve lambda(q, alpha) is called the L-q-bifurcation curve. We show that (i) the unknown nonlinear term is determined as f(u) = u(p) (p > 1) nearly exponentially for u >> 1 under the reasonable condition on lambda(q, alpha) for alpha >> 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.