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2010, Volume 9, Issue 5: 1189-1208. Doi: 10.3934/cpaa.2010.9.1189
This issue Previous Article Kirchhoff systems with nonlinear source and boundary damping terms Next Article The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face

Imperfect bifurcations in nonlinear elliptic equations on spherical caps

  • 1.

    Mathematische Institut, Universät Basel, Rheinsprung 21, CH-4051 Basel

  • 2.

    Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, 599-8531

  • 3.

    Department of Mathematics, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki, 214-8571

Received:  July 31, 2009
Revised:  October 31, 2009
Published:  August 2010
Abstract Related Papers Cited by
    Abstract
  • 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.
    Mathematics Subject Classification: Primary: 35J25; Secondary: 47J25, 58C99.

    Citation:
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