Asymptotic behavior of positive solutions of some quasilinear elliptic problems

2003/12/26 by Zhongmin Guo, Li Ma, Guo, Zhongmin +1
Mathematics · #35B45 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B45

paper · pdf · doi:10.48550/arxiv.math/0312464

25 pages

arxiv created 2003/12/26 · arxiv updated 2009/12/01

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|_{∂ Ω}=0 as q → p-1+0 and as q → ∞ via a scale argument. Here Δ_p is the p-Laplacian with 1<p<∞ 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.

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