A unified approach to Matukuma type equations on the hyperbolic space or on a sphere

Yoshitsugu Kabeya

doi:10.3934/proc.2013.2013.385

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2013, Volume 2013: 385-391. Doi: 10.3934/proc.2013.2013.385 open access

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A unified approach to Matukuma type equations on the hyperbolic space or on a sphere

Yoshitsugu Kabeya^1,

1.
Department of Mathematical Sciences, Osaka Prefecture University, Sakai, 599-8531

Received: September 2012

Revised: February 2013

Published: October 2013

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Abstract

In this note, we consider the following Matukuma type equation on the hyperbolic space or on a spherical cap of the unit sphere under the homogeneous Dirichlet boundary condition \begin{eqnarray*} \Lambda u+K(x)u_+^{p}=0, \end{eqnarray*} where $\Lambda$ is the Laplace-Beltrami operator on the hyperbolic space or on the unit sphere. Under suitable assumptions on $K$, we determine the structure of positive solutions.

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Mathematics Subject Classification: Primary: 35J15; Secondary: 35J61, 58J05.

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References

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