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arXiv · 1009.3102

Coincidence sets in quasilinear elliptic problems of monostable type

Shingo Takeuchi
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Abstract

This paper concerns the formation of a coincidence set for the positive solution of $p$-Laplacian elliptic problems of monostable type. It is proved that for any small parameter of diffusion term, the solution coincides with the stable zero-function $a(x)$ of reaction term in an open set if $a(x)$ is $p$-harmonic (but, not constant) and a zero of order less than 1. Inversely, it is also shown that the solution is less than $a(x)$ if $a(x)$ is a zero of order greater than or equal to 1. The proof rely on comparison theorems and an energy method for obtaining local comaprison functions.

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Shingo Takeuchi. 2010-09-16. Coincidence sets in quasilinear elliptic problems of monostable type. https://doi.org/10.1016/j.jde.2011.07.002

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