Investigation of the global bifurcations/imperfect bifurcations in nonlinear elliptic partial differential equations

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From 2003 to 2005, we investigated on the global bifurcation diagrams on the nonlinear elliptic partial differential equations. Especially, we are concerned with the imperfect bifurcations arising in the Brezis-Nirenberg equation or scalar-field equations with the Robin condition. Due to the Robin condition, we observed several specific properties of the bifurcation diagrams. The diagram has some bending point, which implies the existence of multiple solutions to the equation which we consider. Moreover, we investigated how the bifurcation branch goes to infinity in the function space and we also show the difference between the three dimensional case and the higher dimensional cases. In the three dimensional case, the bending point and the blow-up point move to left in the parameter axis continuously as the parameter in the Robin condition increases. However, if the dimension is not Less than four, then the blow-up point is the origin in the parameter space. We also showed the existence of a threshold value for which the bending occurs. For higher dimensional case, the publishing of the results is not in time, however, the journal acknowledges that our paper will appear in summer 2006.For other co-investigators, they helped me via private communications and encouraged me to research this field although there is no joint work with them.