非周期薛定谔格系统在非扰动和扰动情况下同宿解的研究

The Schrödinger lattice system can be applied in the research from solid state and condensed matter physics to biology, etc. Therefore, it is one of the most important inherently discrete models. In recent years, the study of homoclinic solutions for Schrödinger lattice systems has been widely concerned, but the existed results are all about the non-perturbed case, and most results are about the periodic case. However, by variational methods and critical point theories in combination with tools in nonlinear analysis such as the concentration-compactness principle, the constrained minimization methods and perturbed methods, etc, we will study the existence and multiplicity of homoclinic solutions for non-perturbed and perturbed Schrödinger lattice systems in the non-periodic case. Besides, we also investigate the influence of different perturbed terms and nonlinearities on the numbers and properties for solutions. First, for non-periodic lattice systems with unbounded potentials, we will study the existence and multiplicity of homoclinic solutions in the superlinear, asymptotically linear and sublinear cases. Second, for non-periodic lattice systems with bounded and non-periodic potentials, we will study the existence and multiplicity of homoclinic solutions in the superlinear and asymptotically linear cases. Last, for perturbed and non-periodic lattice systems, we will study the existence and multiplicity of homoclinic solutions in the superlinear, asymptotically linear and sublinear cases. By proposing and solving the problems in the project, we will deepen studies of non-periodic Schrödinger lattice systems and expand applications of this theory in related fields.

薛定谔格系统被广泛应用于固态、凝聚态物理学及生物学等领域，因此它是现有最重要的离散模型之一。近年来，薛定谔格系统同宿解的研究得到了广泛关注，但现有结果都是针对非扰动情况，且大多是针对周期情况。而本项目将在非周期情况下，基于变分法和临界点理论，利用集中紧性原理、限制性极小化方法、扰动方法等非线性分析工具，研究非扰动和扰动的薛定谔格系统同宿解的存在性和多重性，及扰动项和非线性项对解的个数和性质的影响。首先，对无界位势的非周期格系统，研究超线性、渐近线性和次线性的不同情况下同宿解的存在性和多重性；其次，对有界非周期位势的非周期格系统，研究超线性和渐近线性的不同情况下同宿解的存在性和多重性；最后，对扰动的非周期格系统，研究超线性、渐近线性和次线性的不同情况下同宿解的存在性和多重性。本项目中问题的提出和解决，将有助于深化非周期薛定谔格系统的研究，并拓展该理论在相关领域中的应用。

薛定谔格系统被广泛应用于固态、凝聚态物理学及生物学等领域，因此它是现有最重要的离散模型之一。薛定谔格系统同宿解的研究得到了广泛关注，但现有结果都是针对非扰动情况，且大多是针对周期情况。而本项目在非周期情况下，基于变分法和临界点理论，利用集中紧性原理、限制性极小化方法、扰动方法等非线性分析工具，研究了非扰动和扰动的薛定谔格系统同宿解的存在性和多重性，及扰动项和非线性项对解的个数和性质的影响。主要结果如下：1）对无界位势的非周期格系统，得到了超线性、渐近线性和次线性的不同情况下同宿解存在性和多重性的相关结果；2）对有界位势的非周期格系统，得到了超线性和次线性的不同情况下同宿解存在性和多重性的相关结果；3）对扰动的非周期格系统，得到了超线性、渐近线性和次线性的不同情况下同宿解存在性和多重性的相关结果。本项目中问题的提出和解决，将有助于深化非周期薛定谔格系统的研究，并拓展该理论在相关领域中的应用。