### Abstract

This paper deals with singular perturbation problems for quasilin-ear elliptic equations with the p-Laplace operator, e.g., - εδ _{p}u = 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.

Original language | English |
---|---|

Pages (from-to) | 965-973 |

Number of pages | 9 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Issue number | SUPPL. |

Publication status | Published - 2007 Sep |

Externally published | Yes |

### Keywords

- Coincidence set
- Flat core
- P-Laplace operator
- Quasilinear elliptic equation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

**Partial flat core properties associated to the p-Laplace operator.** / Takeuchi, Shingo.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems- Series A*, no. SUPPL., pp. 965-973.

}

TY - JOUR

T1 - Partial flat core properties associated to the p-Laplace operator

AU - Takeuchi, Shingo

PY - 2007/9

Y1 - 2007/9

N2 - This paper deals with singular perturbation problems for quasilin-ear elliptic equations with the p-Laplace operator, e.g., - εδ pu = up-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.

AB - This paper deals with singular perturbation problems for quasilin-ear elliptic equations with the p-Laplace operator, e.g., - εδ pu = up-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.

KW - Coincidence set

KW - Flat core

KW - P-Laplace operator

KW - Quasilinear elliptic equation

UR - http://www.scopus.com/inward/record.url?scp=70349901364&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349901364&partnerID=8YFLogxK

M3 - Article

SP - 965

EP - 973

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - SUPPL.

ER -