Monotonicity Properties of Ground States

Patricio Felmer
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.