### 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 |
---|---|

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 2009 |

### Fingerprint

### 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

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

**Asymptotically critical problems on higher-dimensional spheres.** / Kristály, A.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems*, vol. 23, no. 3, pp. 919-935. https://doi.org/10.3934/dcds.2009.23.919

}

TY - JOUR

T1 - Asymptotically critical problems on higher-dimensional spheres

AU - Kristály, A.

PY - 2009/3

Y1 - 2009/3

N2 - The paper is concerned with the equation -Δhu = f(u) on Sd where Δh denotes the Laplace-Beltrami operator on the standard unit sphere (Sd, 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 H12(Sd) 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 H12-asymptotic behaviour of the sequences of solutions are also fully characterized.

AB - The paper is concerned with the equation -Δhu = f(u) on Sd where Δh denotes the Laplace-Beltrami operator on the standard unit sphere (Sd, 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 H12(Sd) 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 H12-asymptotic behaviour of the sequences of solutions are also fully characterized.

KW - Asymptotically critical growth

KW - Laplacian on the sphere

KW - Non-smooth principle of symmetric criticality

KW - Oscillatory term

KW - Sign-changing solution

KW - Symmetrically distinct solutions

KW - Szulkin-type functional

UR - http://www.scopus.com/inward/record.url?scp=63049139726&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63049139726&partnerID=8YFLogxK

U2 - 10.3934/dcds.2009.23.919

DO - 10.3934/dcds.2009.23.919

M3 - Article

VL - 23

SP - 919

EP - 935

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 3

ER -