Universidad de Chile

Centro de Modelamiento Matematico

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**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

The main result is obtained in collaboration with Quaas, Tang and Ye.