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DC Field | Value | Language |
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dc.contributor.author | Ching, Joshua | - |

dc.date.accessioned | 2017-06-16T04:23:49Z | - |

dc.date.available | 2017-06-16T04:23:49Z | - |

dc.date.issued | 2017-03-30 | - |

dc.identifier.uri | http://hdl.handle.net/2123/16848 | - |

dc.description.abstract | On punctured domains of $\mathbb{R}^N$ with $N \geq 2$, we study non-negative solutions to a nonlinear elliptic equation with a gradient term: $div(|x|^\sigma |\nabla u|^{p-2} \nabla u)=|x|^{-\tau} u^q |\nabla u|^m$ (*), where m,q>0 and 1<p\leq N. In Chapter 1, we review the literature. In Chapter 2, we completely classify the behaviour of solutions near zero for (*) where $\sigma=\tau=0$ and p=2. Suppose that 0<m<2 and m+q>1. Our classification depends on the position of q relative to a critical exponent $q_*=(N−m(N−1))/(N−2)$. We prove the following: If $q<q_*$, then any positive solution u has either (1) a removable singularity at 0, or (2) a weak singularity at 0, or (3) a strong singularity at 0 which is precisely determined. If $q \geq q_*$ (for N>2), then 0 is a removable singularity for all positive solutions. Furthermore, there exist non-constant positive global solutions if and only if $q<q_*$ and in this case, they must be radial, non-increasing with a weak or strong singularity at 0 and converge to any non-negative number at infinity. This is in sharp contrast to the case of m=0 and q>1 when all solutions decay to zero. Our classification theorems are accompanied by corresponding existence results in which we emphasise the more difficult case of 0<m<1 where new phenomena arise. In Chapter 3, we prove gradient bounds for solutions of (*). In particular, we show that if m+q-p+1>0 and $q+1-\sigma-\tau>0$, then the gradient is controlled by a constant multiple of a power of |x|, where the constant is independent of the domain. In Chapter 4, we prove sharp Liouville-type results under minimal regularity assumptions: Let $\tau=0$. The only positive solution of (*) in $\mathbb{R}^N \setminus \{ 0 \}$ are the positive constants if and only if $0<m \leq p$ and m+q-p+1>0. For $\tau \in \mathbb{R}$, we also prove two general Liouville-type theorems. | en_AU |

dc.publisher | University of Sydney | en_AU |

dc.publisher | Faculty of Science | en_AU |

dc.publisher | School of Mathematics and Statistics | en_AU |

dc.rights | The author retains copyright of this thesis. It may only be used for the purposes of research and study. It must not be used for any other purposes and may not be transmitted or shared with others without prior permission. | en_AU |

dc.subject | Liouville | en_AU |

dc.subject | gradient estimate | en_AU |

dc.subject | isolated singularities | en_AU |

dc.subject | PDE | en_AU |

dc.title | Singular Solutions of Nonlinear Elliptic Equations with Gradient Terms | en_AU |

dc.type | PhD Doctorate | en_AU |

dc.type.pubtype | Doctor of Philosophy Ph.D. | en_AU |

dc.description.disclaimer | Access is restricted to staff and students of the University of Sydney . UniKey credentials are required. Non university access may be obtained by visiting the University of Sydney Library. | en_AU |

Appears in Collections: | Sydney Digital Theses (University of Sydney Access only) |

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ching_j_thesis.pdf | Thesis | 1.27 MB | Adobe PDF |

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