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Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞

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Abstract

Here, all solutions of the form u=rkf(φ) to the p-harmonic equation, div(|∇u|p−2∇u)=0, (p>2) in the plane are determined. One main result is a representation formula for such solutions. Further, solutions with an isolated singularity at the origin are constructed (Theorem 1). Graphical illustrations are given at the end of the paper. Finally, all solutions u=rkf(φ) of the limit equation for p=∞, u 2x uxx+2uxuyuxy+u 2y uyy=2, are constructed, some of which have a “strong” singularity at the origin (Theorem 2).

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  1. Department of Mathematics, University of Luleå, S-951 87, Luleå, Sweden

    Gunnar Aronsson

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  1. Gunnar Aronsson
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Aronsson, G. Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞. Manuscripta Math 56, 135–158 (1986). https://doi.org/10.1007/BF01172152

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  • DOI: https://doi.org/10.1007/BF01172152

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