| Summary: | 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.
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