Asymptotic behavior of positive solutions of some quasilinear elliptic problems | Zendy

Guo Zongming | Zendy; Ma Li | Zendy

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Asymptotic behavior of positive solutions of some quasilinear elliptic problems

Author(s) -

Guo Zongming,

Ma Li

Publication year - 2007

Publication title -

journal of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.441

H-Index - 62

eISSN - 1469-7750

pISSN - 0024-6107

DOI - 10.1112/jlms/jdm062

Subject(s) - degenerate energy levels , p laplacian , laplace operator , nonlinear system , argument (complex analysis) , operator (biology) , mathematics , elliptic operator , population , boundary (topology) , pure mathematics , boundary value problem , scale (ratio) , product (mathematics) , mathematical analysis , mathematical physics , physics , chemistry , geometry , quantum mechanics , biochemistry , demography , repressor , sociology , transcription factor , gene

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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