Asymptotic behavior of positive solutions of some quasilinear elliptic problems

Abstract

We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem $-Δ_p u=a u^{p-1}-b(x) u^q$, $u|_{\partial Ω}=0$ as $q \to p-1+0$ and as $q \to \infty$ via a scale argument. Here $Δ_p$ is the $p$-Laplacian with $1<p<\infty$ and $q>p-1$. If $p=2$, such problems arise in population dynamics. Our main results generalize the results for $p=2$, but some technical difficulties arising from the nonlinear degenerate operator $-Δ_p$ are successfully overcome. As a by-product, we can solve a free boundary problem for a nonlinear $p$-Laplacian equation.