Classification of entire solutions of (-Δ)^{𝑁}𝑢+𝑢^{-(4𝑁-1)}=0 with exact linear growth at infinity in 𝐑^{2𝐍-1}

Ngô, Quốc

doi:10.1090/proc/13960

Classification of entire solutions of $(-\Delta )^N u + u^{-(4N-1)}= 0$ with exact linear growth at infinity in $\mathbf {R}^{2N-1}$
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Proc. Amer. Math. Soc. 146 (2018), 2585-2600 Request permission

Abstract:

In this paper, we study global positive $C^{2N}$-solutions of the geometrically interesting equation $(-\Delta )^N u + u^{-(4N-1)}= 0$ in $\mathbf {R}^{2N-1}$. Using the sub poly-harmonic property for positive $C^{2N}$-solutions of the differential inequality $(-\Delta )^N u < 0$ in $\mathbf {R}^{2N-1}$, we prove that any $C^{2N}$-solution $u$ of the equation having linear growth at infinity must satisfy the integral equation \[ u(x) = \int _{\mathbf {R}^{2N-1}} {|x - y|{u^{-(4N-1)}}(y)dy} \] up to a multiple constant and hence take the following form: \[ u(x) = (1+|x|^2)^{1/2} \] in $\mathbf {R}^{2N-1}$ up to dilations and translations. We also provide several non-existence results for positive $C^{2N}$-solutions of $(-\Delta )^N u = u^{-(4N-1)}$ in $\mathbf {R}^{2N-1}$.

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