### Abstract

The paper is concerned with the equation -Δ_{h}u = f(u) on S^{d} where Δ_{h} denotes the Laplace-Beltrami operator on the standard unit sphere (S^{d}, h), while the continuous nonlinearity f : ℝ → ℝ oscillates either at zero or at infinity having an asymptotically critical growth in the Sobolev sense. In both cases, by using a group-theoretical argument and an appropriate variational approach, we establish the existence of [d/2] + (-1)^{d+1} - 1 sequences of sign-changing weak solutions in H_{1}^{2}(S^{d}) whose elements in different sequences are mutually symmetrically distinct whenever f has certain symmetry and d ≥ 5. Although we are dealing with a smooth problem, we are forced to use a non-smooth version of the principle of symmetric criticality (see Kobayashi-Ôtani, J. Funct. Anal. 214 (2004), 428-449). The L ^{∞}- and H_{1}^{2}-asymptotic behaviour of the sequences of solutions are also fully characterized.

Original language | English |
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Pages (from-to) | 919-935 |

Number of pages | 17 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 23 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 1 2009 |

### Keywords

- Asymptotically critical growth
- Laplacian on the sphere
- Non-smooth principle of symmetric criticality
- Oscillatory term
- Sign-changing solution
- Symmetrically distinct solutions
- Szulkin-type functional

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics