【摘要】 利用变分方法研究了非线性离散共振Schrdinger系统非平凡解的存在性.首先将该系统转化为矩阵形式,给出了它所对应的能量泛函,于是该系统的解等价于能量泛函的临界点.同时,利用矩阵张量积的特征值性质得到了该系统所对应的线性特征值系统的全部特征值,从而得到了系统共振现象的数学描述.进一步,当系统在零点或无穷远点发生共振时,在一定的假设条件下,通过临界群的计算,结合Morse理论,证明了此系统至少存在一个非平凡解.更多还原

【Abstract】 The existence of nontrivial solutions for a nonlinear discrete resonant Schrdinger systems was studied by using variational methods. First,the matrix form and the corresponding energy function of this system were given. Then,the desired solutions were equivalent to the critical points of the energy function. Meanwhile,by using the properties of eigenvalues of matrices tensor products,all the eigenvalues of the systems corresponding to the linear systems were obtained. As a result,the mathematical descriptions of the resonant phenomena of this system are given.And then when the systems is resonant at zero or at infinity,it is proved that there exists at least one nontrivial solution for the systems under certain assumptions via computations of the critical groups and Morse theory.更多还原