Partial flat core properties associated to the $p$-laplace operator
Shingo Takeuchi · 2007
Abstract. This paper deals with singular perturbation problems for quasilinear elliptic equations with the p-Laplace operator, e.g., −ε∆pu = u p−1 |a(x) − u | q−1 (a(x) − u), where ε 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 ε → 0. In particular, when q < p−1 and ε 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. 1. Introduction. In