Differential and Integral Equations

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 and Quôc Anh Ngô

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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.

Article information

Source
Differential Integral Equations, Volume 30, Number 11/12 (2017), 917-928.

Dates
First available in Project Euclid: 1 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.die/1504231279

Mathematical Reviews number (MathSciNet)
MR3693991

Zentralblatt MATH identifier
06819584

Subjects
Primary: 35B45: A priori estimates 35J40: Boundary value problems for higher-order elliptic equations 35J60: Nonlinear elliptic equations

Citation

Duoc, Trinh Viet; Ngô, Quôc Anh. 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 (2017), no. 11/12, 917--928. https://projecteuclid.org/euclid.die/1504231279


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