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

Abstract

We consider the semilinear elliptic eigenvalue problem −Δu + k(|x|)up = λu, u > 0 in BR, u = 0 on ∂BR, where p > 1 is a constant, BR : = {x ∈ RN : |x| < R}(N ≥ 1), and λ > 0 is a parameter. We investigate the global structure of the branch of (λ, uλ) of bifurcation diagram from a point of view of L2‐theory. To do this, we establish a precise asymptotic formula for λ = λ(α) as α → ∞, where .