Asymptotic behavior of positive solutions of some quasilinear elliptic problems

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