Partial flat core properties associated to the $p$-laplace operator

Pages: 965 - 973, Issue Special, September 2007

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Shingo Takeuchi - Kogakuin University, 2665-1 Nakano, Hachioji, Tokyo 192-0015, Japan (email)

Abstract: This paper deals with singular perturbation problems for quasilinear elliptic equations with the $p$-Laplace operator, e.g., −$\epsilon_pu = u^(p − 1)|a(x) − u|^(q − 1)(a(x) − u)$, where $\Epsilon$ is a positive parameter, $p$ > 1, $q$ > 0 and $a(x)$ is a positive continuous function. It is proved that any positive solution converges to $a(x)$ uniformly in any compact subset as $\epsilon \rightarrow 0$. In particular, when $q$ < $p$−1 and $\epsilon$ is small enough, the solutions coincide with $a(x)$ on one or more than one subdomain where $a(x)$ is constant, and hence there appear flat cores partially in the whole domain. These results are proved by comparison principles.

Keywords:  Coincidence set, flat core, p-Laplace operator, quasilinear elliptic equation.
Mathematics Subject Classification:  Primary: 35J70; Secondary: 35J25.

Received: September 2006;      Revised: August 2007;      Published: September 2007.