## Abstract

We study the leading order behaviour of positive solutions of the equation -Δ u+εu-2u C |u|^{p-2}u+|u|^{q-2}u 0, xε ℝ^{N} where N ≥ 3, q > p > 2 and ε > 0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p, q and N. The behaviour of solutions depends on whether p is less than, equal to or greater than the critical Sobolev exponent 2* = 2N/N-2 . For p < 2* the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p > 2* the solution asymptotically coincides with the solution of the equation with ε = 0. In the most delicate case p = 2* the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden-Fowler equation, whose choice depends on ε in a nontrivial way.

Original language | English (US) |
---|---|

Pages (from-to) | 1081-1109 |

Number of pages | 29 |

Journal | Journal of the European Mathematical Society |

Volume | 16 |

Issue number | 5 |

DOIs | |

State | Published - 2014 |

## All Science Journal Classification (ASJC) codes

- General Mathematics
- Applied Mathematics

## Keywords

- Asymptotic behaviour
- Critical Sobolev exponent
- Critical and supercritical nonlinearity
- Pohožaev identity
- Subcritical