Asymptotic behavior of positive solutions of some quasilinear elliptic problems

We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem $-\Delta_p u=a u^{p-1}-b(x) u^q$, $u|_{\partial \Omega}=0$ as $q \to p-1+0$ and as $q \to \infty$ via a scale argument. Here $\Delta_p$ 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 $-\Delta_p$ are successfully overcome. As a by-product, we can solve a free boundary problem for a nonlinear $p$-Laplacian equation.