J. Ildefonso Diaz, Monica Lazzo, Paul G. Schmidt
This paper concerns the equation , where , , and denotes the Laplace operator in , for some . Specifically, we are interested in the structure of the set of all large radial solutions on the open unit ball in . In the well-understood second-order case, the set consists of exactly two solutions if the equation is subcritical, of exactly one solution if it is critical or supercritical. In the fourth-order case, we show that is homeomorphic to the unit circle if the equation is subcritical, to minus a single point if it is critical or supercritical. For arbitrary , the set is a full -sphere whenever the equation is subcritical. We conjecture, but have not been able to prove in general, that is a punctured -sphere whenever the equation is critical or supercritical. These results and the conjecture are closely related to the existence and uniqueness (up to scaling) of entire radial solutions. Understanding the geometric and topological structure of the set allows precise statements about the existence and multiplicity of large radial solutions with prescribed center values .
Published May 15, 2007.
Math Subject Classifications: 35J40, 35J60.
Key Words: Polyharmonic equation; radial solutions; entire solutions; large solutions; existence and multiplicity; boundary blow-up.
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| Jesus Ildefonso Díaz |
Departamento de Matemática Aplicada
Universidad Complutense de Madrid, 28040 Madrid, Spain
| Monica Lazzo |
Dipartimento di Matematica, Universitá di Bari
via Orabona 4, 70125 Bari, Italy
| Paul G. Schmidt |
Department of Mathematics and Statistics
Auburn University, AL 36849-5310, USA
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