### Abstract

Prescribing σ_{k} curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. For a given a positive function K to be prescribed on the 4-dimensional round sphere, we obtain asymptotic profile analysis for potentially blowing up solutions to the σ_{2} curvature equation with the given K; and rule out the possibility of blowing up solutions when K satisfies a non-degeneracy condition. Under the same non-degeneracy condition on K, we also prove uniform a priori estimates for solutions to a family of σ_{2} curvature equations deforming K to a positive constant; and under an additional, natural degree condition on a finite dimensional map associated with K, we prove the existence of a solution to the σ_{2} curvature equation with the given K using a degree argument involving fully nonlinear elliptic operators to the above deformation.

Original language | English (US) |
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Pages (from-to) | 539-565 |

Number of pages | 27 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 40 |

Issue number | 3 |

DOIs | |

State | Published - Jan 24 2011 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics