The Quantitative Analysis of Singular Solutions for Semilinear Elliptic Equations with Nonlinear Critical and Supercritical Potential

• Main Authors: Chih-Her Chen, 陳志和
• Other Authors: Jann-Long Chern
• Format: Others
• Language: en_US
• Published: 2018
• Online Access: http://ndltd.ncl.edu.tw/handle/fxurd7

博士 === 國立中央大學 === 數學系 === 106 === For the first part, we consider the structure of singular solutions for elliptic equations with the Hardy potential and critical nonlinearity under quite general conditions on the potential terms. In general, it is shown that there exists a unique special singular solution, and other infinitely many singular solutions are oscillatory around the special singular solution. We also study the asymptotic behavior of the solutions around the singular point. Our results can be applied to various problems such as the scalar field equation, a self-replication model and the Cafarelli-Kohn- Nirenberg inequality. In particular, we discuss the three elliptic equations separately and to consider the asymptotic behavior of the solutions at infinity under supercritical case or classify all the solutions structure according to the characteristic of each equation. For the second part, we prove the existence of Non-Topological solutions for the elliptic system arising from a product Abelian Gauge Field theory.