Monotonicity properties for ground states of the scalar field equation

• Published in: Annales de l'Institut Henri Poincaré. Analyse non linéaire Vol. 25; no. 1; pp. 105 - 119
• Main Authors: Felmer, Patricio L., Quaas, Alexander, Tang, Moxun, Yu, Jianshe
• Format: Journal Article
• Language: English
• Published: Paris Elsevier Masson SAS 01.01.2008; Elsevier
• Subjects: Exact sciences and technology; Ground states; Liouville type theorem; Mathematical analysis; Mathematics; Maximum value; Monotonic property; Sciences and techniques of general use; Uniqueness and existence; secondary; Ground states; Maximum value; Monotonic property; Liouville type theorem; Uniqueness and existence; primary; Monotonic function; Liouville theorem; Increasing function; Nonlinear analysis; Positive solution; Scalar field; Ground state; Dirichlet problem; Mathematics; Monotonicity; Field equation
• ISSN: 0294-1449; 1873-1430
• DOI: 10.1016/j.anihpc.2006.12.003

It is well known that the scalar field equationΔu−u+up=0inRN,N⩾3, 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‖∞, 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 eN/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).