临界群与离散共振广义Emden-Fowler方程解的多重性

Multiplicity of Solutions for a Discrete Generalized Emden-Fowler Equation with Resonance Via Computations of the Critical Groups

针对离散广义Emden-Fowler方程在零点与无穷远点同时共振于零特征值的情形时,多个非平凡解的存在性问题,首先将其转化为矩阵形式,同时给出了相应的能量泛函,进而利用变分方法,将该问题的解等价于能量泛函的临界点.当非线性项在零点及无穷远点满足一定的假设条件时,结合Morse理论,通过临界群的计算,分别证明了此问题至少存在一个及两个非平凡解.所得结论丰富了离散广义Emden-Fowler方程的研究结果,对其它离散非线性问题的同类研究工作也有一定的指导意义.

The existence problem of multiple nontrivial solutions for a discrete generalized Emden-Fowler equation which is resonant at zero eigenvalue both at zero and infinity simultaneously was studied. The matrix form and the corresponding energy functional of the above problem were given, then the desired solutions were equivalent to the critical points of the energy functional by variational methods. When the nonlinear term met certain assumptions at zero and infinity, it was proved that the problem had at least one and two nontrivi- al solutions respectively via Morse theory and computations of the critical groups. The conclusion enriches re- search results of discrete generalized Emden-Fowler equations, which is significant for the further research on other discrete nonlinear problems.