Date of Award:

5-2000

Document Type:

Dissertation

Degree Name:

Doctor of Philosophy (PhD)

Department:

Mathematics and Statistics

Advisor/Chair:

Zhi-Qiang Wang

Co-Advisor/Chair:

Ian Anderson

Third Advisor:

David Sattinger

Abstract

In this dissertation, we establish existence and multiplicity of positive solutions for semilinear elliptic equations with subcritical and critical nonlinearities. We treat problems invariant under subgroups of the orthogonal group. Roughly speaking, we prove that if enough "mass " is concentrated around special orbits, then among the functions with prescribed symmetry, there is a solution for the original problem.

Our results can be regarded as a further development of the work of Z.-Q. Wang, where existence of local minima in the space of symmetric functions was studied for the Schrödinger equation. We illustrate the general theory with three examples, all of which produce new results. Our method allows the construction of solutions with prescribed symmetry, and it represents a step further in the classification of positive solutions for certain nonlinear elliptic problems.

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Included in

Mathematics Commons

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