非线性Schrodinger-Poisson方程组的驻波解是近10年来非常活跃的研究课题. 大部分这方面的研究只限于Schrodinger算子是正算子的情形, 这时对应的能量泛函有山路几何结构, 人们用山路引理等方法得到许多重要的结果. 然而, 如果要寻求原时变Schrodinger-Poisson方程组的高频驻波解, 则相应的稳态方程组中的Schrodinger算子不再是正算子, 对于这种情形, 结果非常少. 本项目将在我们近期在这方面的研究工作的基础上, 系统地研究Schrodinger算子有负空间的不定情形的非线性Schrodinger-Poisson方程组的解的存在性和多重性. 我们将考虑包括周期位势在内的强不定情形. 对于非线性项, 我们将考虑渐近线性, 4-超线性的情形.

The standing wave of nonlinear Schrodinger-Poisson systems is a very active area of research in the last 10 years. Most research on this topic are limited on the case that the Schrodinger operator is positive. In this case the energy functional has the mountain pass geometry and lots of interesting results are obtained via the mountain pass lemma. However, if we are interested in high frequency standing waves of the time dependent Schrodinger-Poisson systems, the Schrodinger operator in the stationary system is not positive. For this case there are very few results. In this project, base on our recent research in this area, we will systematically study the existence and multiplicity of standing waves nonlinear Schrodinger-Poisson systems in the case that the Schrodinger operator is indefinite. For the potential, our consideration includes the strongly indefinite case with periodic potential, for the nonlinearity, we will consider the asymptotically linear case as well as the 4-superlinear case.