Monotonicity Properties of Ground States
Patricio Felmer
Universidad de Chile
Centro de Modelamiento Matematico
email: pfelmer@dim.uchile.cl
Abstract: It has been well-known that the scalar field
equation
admits ground state solutions if and only if
. For each fixed
in this range, there corresponds a unique ground state (up to
translation), as well as a unique positive solution to the Dirichlet boundary
value problem
for any finite ball in . In this talk we show that the maximum value
of such ground states, , is an increasing function of for
all
, with
, and
as
. Consequently, we derive a Liouville type
theorem ensuring that there exists neither a ground state solution to this
equation, nor a positive solution of the Dirichlet problem in any finite ball,
with the maximum value less than . Our proof relies on some fine
analyses on the first variation of ground states with respect to . The
delicacy of this study can be evidenced by the fact that, on any fixed finite
ball, the maximum value of positive solutions to the Dirichlet problem is never a monotone function of over the whole range
. We will also present some related problems and open
questions.
The main result is obtained in collaboration with Quaas, Tang and Ye.