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

2003-2005年间，我们研究了一类非线性椭圆型偏微分方程解的整体分岔图。特别地，我们考虑了具有Robin条件的Brezis-Nirenberg方程或标量场方程解的不完全分叉。由于Robin条件，我们观察到了分岔图的几个特殊性质。这张图有一些弯曲点，这意味着我们所考虑的方程存在多个解。此外，我们还研究了分支分支在函数空间中如何到达无穷远，并指出了三维情形与高维情形的区别。在三维情况下，随着Robin条件下参数的增加，弯曲点和爆破点在参数轴上不断左移。然而，如果维度不小于4，则爆破点是参数空间中的原点。我们还证明了发生弯曲的阈值的存在。对于更高维度的案例，结果的发表并不及时，但该杂志承认，我们的论文将于2006年夏季发表。对于其他合作调查人员，他们通过私下沟通帮助我，并鼓励我研究这一领域，尽管没有与他们合作。

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.