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

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.