Imperfect bifurcations in nonlinear elliptic equations on spherical caps

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September 2010, 9(5): 1189-1208. doi: 10.3934/cpaa.2010.9.1189

Imperfect bifurcations in nonlinear elliptic equations on spherical caps

C. Bandle ^1, , Y. Kabeya ^2, and Hirokazu Ninomiya ^3,

1.	Mathematische Institut, Universät Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
2.	Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, 599-8531, Japan
3.	Department of Mathematics, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki, 214-8571, Japan

Received August 2009 Revised November 2009 Published May 2010

Abstract
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We consider elliptic boundary value problems on large spherical caps with parameter dependent power nonlinearities. In this paper we show that imperfect bifurcation occurs as in the work [13]. When the domain is the whole sphere, there is a constant solution. In the case where the domain is a spherical cap, however, the constant solution disappears due to the boundary condition. For large spherical caps we construct solutions which are close to the constant solution in the whole n-dimensional sphere, using the eigenvalues of the linearized problem in the whole sphere and fixed point arguments based on a Lyapunov-Schmidt type reduction. Numerically there is a surprising similarity between the diagrams of this problem and the ones obtained in [18], also [5], for a Brezis-Nirenberg type problem on spherical caps.

Keywords: Nonlinear elliptic equations, spherical coordinates, fixed point theorems., bifurcation.

Mathematics Subject Classification: Primary: 35J25; Secondary: 47J25, 58C9.

Citation: C. Bandle, Y. Kabeya, Hirokazu Ninomiya. Imperfect bifurcations in nonlinear elliptic equations on spherical caps. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1189-1208. doi: 10.3934/cpaa.2010.9.1189

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