Precise spectral asymptotics for nonautonomous logistic equations of population dynamics in a ball

We consider the semilinear elliptic eigenvalue problem -Delta u+k(vertical bar x vertical bar)u(p) = lambda u, u > 0 in B-R, u = 0 on partial derivative B-R, where p > 1 is a constant, B-R := {x is an element of R-N : vertical bar x vertical bar < R} (N = 1), and lambda > 0 is a parameter. We investigate the global structure of the branch of (lambda,u(lambda)) of bifurcation diagram from a point of view of L-2-theory. To do this, we establish a precise asymptotic formula for lambda = lambda(a) as alpha -> infinity, where alpha := vertical bar vertical bar u lambda vertical bar vertical bar(2).