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

Pages: 385 - 391, Issue special, November 2013

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Yoshitsugu Kabeya - Department of Mathematical Sciences, Osaka Prefecture University, Sakai, 599-8531, Japan (email)

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.

Keywords:  Matukuma type equations, hyperbolic space, spherical cap.
Mathematics Subject Classification:  Primary: 35J15; Secondary: 35J61, 58J05.

Received: September 2012;      Revised: February 2013;      Published: November 2013.