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|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 α:=‖uλ‖2.