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November/December 2017 A note on positive radial solutions of $\Delta^2 u + u^{-q} = 0$ in $\mathbf R^3$ with exactly quadratic growth at infinity
Trinh Viet Duoc, Quôc Anh Ngô
Differential Integral Equations 30(11/12): 917-928 (November/December 2017).

Abstract

Of interest in this note is the following geometric equation, $\Delta^2 u + u^{-q} = 0$ in $\mathbf R^3$. It was found by Choi--Xu (J. Differential Equations, 246, 216–234) and McKenna–Reichel (Electron. J. Differential Equations, 37 (2003)) 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, Guerra (J. Differential Equations, {253}, 3147–3157) 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.

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Trinh Viet Duoc. Quôc Anh Ngô. "A note on positive radial solutions of $\Delta^2 u + u^{-q} = 0$ in $\mathbf R^3$ with exactly quadratic growth at infinity." Differential Integral Equations 30 (11/12) 917 - 928, November/December 2017.

Information

Published: November/December 2017
First available in Project Euclid: 1 September 2017

zbMATH: 06819584
MathSciNet: MR3693991

Subjects:
Primary: 35B45, 35J40, 35J60

Rights: Copyright © 2017 Khayyam Publishing, Inc.

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Vol.30 • No. 11/12 • November/December 2017
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