Imperfect bifurcations in nonlinearelliptic equations on spherical caps
Catherine Bandle, Yoshitsugu Kabeya, Hirokazu Ninomiya · Communications on Pure & Applied Analysis · 2010
We consider elliptic boundary value problems on large sphericalcaps with parameter dependent power nonlinearities. In this paper we showthat imperfect bifurcation occurs as in the work [13]. When the domain is thewhole sphere, there is a constant solution. In the case where the domain is aspherical cap, however, the constant solution disappears due to the boundarycondition. For large spherical caps we construct solutions which are close to theconstant solution in the whole n-dimensional sphere, using the eigenvalues ofthe linearized problem in the whole sphere and fixed point arguments based on aLyapunov-Schmidt type reduction. Numerically there is a surprising similaritybetween the diagrams of this problem and the ones obtained in [18], also [5],for a Brezis-Nirenberg type problem on spherical caps.