2006 International Conference in honor of Jacqueline Fleckinger.
Electron. J. Diff. Eqns., Conference 16 (2007), pp. 103128.
Large radial solutions of a polyharmonic equation
with superlinear growth
J. Ildefonso Diaz, Monica Lazzo, Paul G. Schmidt
Abstract:
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 wellunderstood secondorder 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 fourthorder 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 blowup.
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Jesus Ildefonso Díaz
Departamento de Matemática Aplicada
Universidad Complutense de Madrid,
28040 Madrid, Spain
email: ji_diaz@mat.ucm.es


Monica Lazzo
Dipartimento di Matematica, Universitá di Bari
via Orabona 4, 70125 Bari, Italy
email: lazzo@dm.uniba.it 

Paul G. Schmidt
Department of Mathematics and Statistics
Auburn University, AL 368495310, USA
email: pgs@auburn.edu 
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