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Journal of the London Mathematical Society Advance Access originally published online on October 10, 2007
Journal of the London Mathematical Society 2007 76(2):419-437; doi:10.1112/jlms/jdm062
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© 2007 London Mathematical Society

Asymptotic behavior of positive solutions of some quasilinear elliptic problems

Zongming Guo and Li Ma

Department of Mathematics
Dong Hua University
Shanghai 200051
P.R. China
guozm{at}public.xxptt.ha.cn
Department of Mathematical Sciences
Tsinghua University
Beijing 100084
P.R. China

We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem –{Delta}pu = a up–1–b(x) uq, u|{partial} {Omega} = 0, as q -> p – 1 + 0 and as q -> {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.

2000 Mathematics Subject Classification 35B45 (primary), 35J40 (secondary).

Received May 31, 2006; published online October 10, 2007.


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