Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations

By Yihong Du, Zongming Guo. Advances in Differential Equations, 2002

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. For a wide class of nonlinearities f ( u ) satisfying f (0) = f ( a ) = 0, f ( u ) > 0 in (0 , a ) and f ( u ) < 0 in ( a, ∞ ), we study the quasilinear equation − Δ p u = λc ( x ) f ( u ) over the entire R N or over a bounded smooth domain Ω. Such equation covers various models from chemical reaction theory and population biology. We show that any nonnegative solution on the entire R N must be a constant, and for large λ , any positive solution on Ω must approach a in compact subsets of Ω, no matter whether or not the solution has a prescribed behavior near the boundary of Ω. We also determine the flat cores of the positive solutions and show that the flat cores enlarge to the whole Ω as λ goes to infinity. Our proof of these results demonstrates the usefulness of boundary blow-up solutions in various classical problems.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c7c01c7610753fbc184553535952ae4910d02e91

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In Applied Mathematics

  1. Lishan Liu
  2. Yonghong Wu
  3. Ahmed Mohammed
  4. Xinguang Zhang
  5. Yujun Cui
  6. Zhijun Zhang

In Computational Theory and Mathematics

  1. Ahmed Mohammed
  2. Lishan Liu
  3. Yonghong Wu

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