Asymptotic behaviour of $L^q$-norm of solutions to elliptic eigenvalue problems arising in population dynamics

Abstract

We consider the nonlinear elliptic eigenvalue problem $-\Delta u+u^p=\lambda u$ in $B_R$, $u>0$ in $B_R$, $u=0$ on $\partial B_R$, where $B_R:=\{|x|<R\}\subset\mathbb{R}^N$ ($N\ge2$) and $p>1$ is a constant. This equation is well known as a model equation of population density for some species when $p=2$. Here, $\lambda>0$ represents the reciprocal number of its diffusion rate and $\Vert u\Vert_1$ stands for the mass of the species. We establish the precise asymptotic formula for $\Vert u_\lambda\Vert_q$ as $\lambda\to\infty$, where $1\le q<\infty$. We also obtain the difference between $\Vert u_\lambda\Vert_\infty$ and $\Vert u_\lambda\Vert_q$ when $\lambda\gg1$.