Monotonicity properties for ground states of the scalar field equation

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, $\Vert u{\Vert }_{\infty }$, 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 $1<p<(N+2)/(N-2)$.