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Non-radial solutions to a bi-harmonic equation with negative exponent

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Abstract

We prove the existence of smooth non-radial entire solution to

$$\begin{aligned} \Delta ^2 u+u^{-q}=0\quad \text {in }\mathbb {R}^3,\quad u>0, \end{aligned}$$

for \(q>1\). This answers an open question raised by McKenna and Reichel (Electron J Differ Equ 37:1–3, 2003).

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T1Z2, Canada

    Ali Hyder & Juncheng Wei

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  1. Ali Hyder
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  2. Juncheng Wei
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Correspondence to Ali Hyder.

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Communicated by M. Struwe.

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Ali Hyder: The author is supported by the Swiss National Science Foundation, Grant No. P2BSP2-172064. Juncheng Wei: The research is partially supported by NSERC.

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Hyder, A., Wei, J. Non-radial solutions to a bi-harmonic equation with negative exponent. Calc. Var. 58, 198 (2019). https://doi.org/10.1007/s00526-019-1647-4

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