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

Abstract

We consider the semilinear elliptic eigenvalue problem $-\Delta u+k\left(\left|x\right|\right)u^p=\lambda u$, $u>0$ in $B_R$, $u=0$ on $\partial B_R$, where $p>1$ is a constant, , and $\lambda >0$ is a parameter. We investigate the global structure of the branch of $\left(\lambda ,u_{\lambda }\right)$ of bifurcation diagram from a point of view of $L^2$-theory. To do this, we establish a precise asymptotic formula for $\lambda =\lambda \left(\alpha \right)$ as $\alpha \to \infty$, where $\alpha :={\Vert u_{\lambda }\Vert }_2$.