Asymptotic behavior of positive solutions of some quasilinear elliptic problems

Details Asymptotic behavior of positive solutions of some quasilinear elliptic problems Asymptotic behavior of positive solutions of some quasilinear elliptic problems

2003-12-26

arxiv.org/abs/math/0312464v1

Mathematics

Analysis of PDEs

25 pages

Scientific paper

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 $1p-1$. 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.

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