# Monotonicity properties for ground states of the scalar field equation

### Patricio L. Felmer

Departamento de Ing. Matemática, F.C.F.M. Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile### Alexander Quaas

Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110, Avda. España 1680, Valparaíso, Chile### Moxun Tang

Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA### Jianshe Yu

College of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, PR China

## Abstract

It is well known that the scalar field equation

admits ground state solutions if and only if $1 < p < (N + 2)/ (N−2)$ and that for each fixed *p* in this range, there corresponds a unique ground state (up to translation). In this article, we show that the maximum value of such ground states, \|u\|_{\infty }\right., is an increasing function of *p* for all $1 < p < (N + 2)/ (N−2)$. As a consequence of this result 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 $\mathrm{e}^{N/ 4}$. Our proof relies on some fine analyses on the first variation of ground states with respect to the initial value and with respect to *p*. 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 *p*, over the whole range $1 < p < (N + 2)/ (N−2)$.

## Cite this article

Patricio L. Felmer, Alexander Quaas, Moxun Tang, Jianshe Yu, Monotonicity properties for ground states of the scalar field equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 1, pp. 105–119

DOI 10.1016/J.ANIHPC.2006.12.003