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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

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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