We studied some semilinear elliptic equations on the entire space R^N. Our approach was variational, and the major obstacle was the breakdown in compactness due to the unboundedness of the domain. First, we considered an asymptotically linear Scltrodinger equation under the presence of a steep potential well. Using Lusternik-Schnirelmann theory, we obtained multiple solutions depending on the interplay between the linear, and nonlinear parts. We also exploited the nodal structure of the solutions. For periodic potentials, we constructed infinitely many homoclinic-type multibump solutions. This recovers the analogues result for the superlinear case. Finally, we introduced weights on the linear and nonlinear parts, and studied how their interact ion affects the local and global compactness of the problem. Our approach is based on the Caffarelli-Kohn-Nirenberg inequalities.
| Identifer | oai:union.ndltd.org:UTAHS/oai:digitalcommons.usu.edu:etd-8243 |
| Date | 01 May 2004 |
| Creators | van Heerden, Francois A. |
| Publisher | DigitalCommons@USU |
| Source Sets | Utah State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | All Graduate Theses and Dissertations |
| Rights | Copyright for this work is held by the author. Transmission or reproduction of materials protected by copyright beyond that allowed by fair use requires the written permission of the copyright owners. Works not in the public domain cannot be commercially exploited without permission of the copyright owner. Responsibility for any use rests exclusively with the user. For more information contact digitalcommons@usu.edu. |
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