二次谐波方程组的解

Recently, nonlinear elliptic systems, especially Schrödinger systems, have been studied extensively. Since the elliptic systems studied in the project are related to second harmonic generation phenomena in optic, we call them the second harmonic generation systems. By the sign of the parameters in the systems, we divide the second harmonic generation systems into three parts: I, II and III. The existence of solutions and the boundedness of solutions to I systems have been studied respectively by using critical point theory and cut-off technique which maybe do not work in the study of II and III systems. In this project, the second harmonic generation system, especially II and III systems, will be investigated extensively. By using the appropriate nonlinear methods, such as bifurcation theory, the existence of solutions of the systems will be studied as well as local bifurcation phenomena, global bifurcation phenomena and uniqueness of positive solutions.

近年来，非线性椭圆方程组尤其是薛定谔型方程组的研究受到了广泛关注。本项目研究的非线性椭圆方程组与光学中的二次谐波现象有关，我们把它称为二次谐波方程组。根据二次谐波方程组中参数正负号的不同我们又把二次谐波方程组分成I型、II型和III型方程组。对于I型方程组，利用临界点理论可讨论方程组解的存在性，利用截断技巧可以讨论解的有界性。但是，这些方法并不适用于II型与III型方程组。本项目对二次谐波方程组，特别是II型与III型方程组进行深入研究。通过有效利用非线性泛函分析方法，例如分歧理论，研究二次谐波方程组的解的存在性、局部分歧现象、整体分歧现象以及正解的唯一性问题。

光学二次谐波现象是上世纪六十年代发现的。本项目研究与二次谐波现象有关的一类非线性椭圆方程组。我们称它为二次谐波方程组。利用多种非线性分析方法和理论，我们得到了有界区域和无界区域情形下二次谐波方程组非平凡解存在的充分条件。同时，本项目还讨论了与二次谐波方程组有关的其他方程或方程组解的存在性。项目达到了预期目标，获得了较好的效果，提升了项目组成员的科研水平，加深了对二次谐波方程组及相关问题的认识，为后续进一步研究打下了坚实的基础。