Comprehensive Exam: Ze Cheng
Qualitative Analysis of Some Nonlinear PDE systems
Ze Cheng
Applied Mathematics,听
Date and time:听
Wednesday, December 11, 2013 - 3:00pm
尝辞肠补迟颈辞苍:听
ECOT 831
础产蝉迟谤补肠迟:听
The Hardy-Littlewood-Sobolev (HLS) type system,
\[ \begin{cases} 听听听听听 (-\bigtriangleup)^{k} u = v^q, \,\,u > 0,\,\, \mbox{ in } R^n,\\ 听听听听听 (-\bigtriangleup)^{k} v = u^p, \,\,v > 0,\,\, \mbox{ in } R^n \\ \end{cases} \] is a fundamental nonlinear PDE system, and often arises as the 鈥渂lown-up鈥 equations for many nonlinear problems with the 鈥渁ssociated dominating nonlinearity.鈥 Together with its special cases, this system is perhaps the most studied PDE system in recent decades. It plays crucial roles in geometric analysis, dynamics analysis of vacuum states, study of nonlinear Schr枚dinger equations, and many other research areas.
The system can be categorized into three cases depending on parameters \(p\), \(q\) and \(k\), namely, subcritical case \(\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-2k}{n}\), critical case \(\frac{1}{p+1}+\frac{1}{q+1}=\frac{n-2k}{n}\) and supercritical case \(\frac{1}{p+1}+\frac{1}{q+1}<\frac{n-2k}{n}\). We propose to study that, the non-existence of positive solution of subcritical cases (i.e. the Lane-Emden Conjecture), the essential uniqueness of solution in the critical cases and the existence and uniqueness of solution in the supercritical cases.
By implementing our new or improved methods in analysis, we hope to obtain a deeper understanding and hence make a small step forward to solving the above fundamental problems, which also may lead to better understandings of various PDE problems arising from different research areas in science and engineering.
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