A note on positive radial solutions of \(\Delta^2u+u^{-q}=0\) in \(\mathbf{R}^3\) with exactly quadratic growth at infinity. (English) Zbl 1413.35100

Duoc, Trinh Viet; Ngô, Quôc Anh

Differ. Integral Equ. 30, No. 11-12, 917-928 (2017).

Summary: Of interest in this note is the following geometric equation, \(\Delta^2u+u^{-q}=0\) in \(\mathbf{R}^3\). It was found by Y. S. Choi and X. Xu [J. Differ. Equations 246, No. 1, 216–234 (2009; Zbl 1165.35014)] and P. J. McKenna and W. Reichel [Electron. J. Differ. Equ. 2003, Paper No. 37, 13 p. (2003; Zbl 1109.35321)] that the condition \(q>1\) is necessary and any positive radially symmetric solution grows at least linearly and at most quadratically at infinity for any \(q>1\). In addition, when \(q>3\) any positive radially symmetric solution is either exactly linear growth or exactly quadratic growth at infinity. Recently, I. Guerra [J. Differ. Equations 253, No. 11, 3147–3157 (2012; Zbl 1262.35098)] has shown that the equation always admits a unique positive radially symmetric solution of exactly given linear growth at infinity for any \(q>3\) which is also necessary. In this note, by using the phase-space analysis, we show the existence of infinitely many positive radially symmetric solutions of exactly given quadratic growth at infinity for any \(q>1\), hence completing the picture of positive radially symmetric solutions of the equation.

MSC:

35B45	A priori estimates in context of PDEs
35J40	Boundary value problems for higher-order elliptic equations
35J60	Nonlinear elliptic equations

Keywords:

radially symmetric solution; asymptotic behavior; flat metric

Citations:

Zbl 1165.35014; Zbl 1109.35321; Zbl 1262.35098