广义拟线性薛定谔方程解的存在性及其性态研究

The generalized quasilinear Schrödinger equations which have practical application backgrounds, play an important role in fluid mechanics, plasma physics, dissipative quantum mechanics and condensed matter theory. The study on the existence and properties of solutions for generalized quasilinear Schrödinger equations not only has a theoretical significance, but also has extensive applications. This project aims at studying the existence and properties of solutions for this class of equations. Firstly, we will study the existence, boundedness and concentration of solutions for generalized quasilinear Schrödinger equations with bounded potentials and steep potential wells by using the perturbation methods and minimax methods, respectively. Secondly, the existence and multiplicity of solutions for generalized quasilinear Schrödinger equations with sublinear or asymptotically linear terms will be studied by using Morse theory incorporated with minimax methods. Thirdly, we shall establish a theorem of multiple critical points by combining variational methods with the method of upper and lower solutions, and study the existence and multiplicity of solutions for generalized quasilinear Schrödinger equations with perturbation terms and singular terms. The study of this project will enrich and develop the variational methods and the theory of quasilinear Schrödinger equations, while prompting the development of related fields such as differential equations and dynamic systems.

广义拟线性薛定谔方程是一类具有实际应用背景的微分方程，在流体力学、等离子物理、耗散量子力学和凝聚态理论中有着重要的作用。关于广义拟线性薛定谔方程解的存在性及性态研究不仅具有理论意义而且有广泛的应用价值。本项目主要研究该类方程解的存在性及性态，具体研究内容包括：1)分别运用扰动方法和极小极大原理研究具有有界位势和位势井函数的广义拟线性薛定谔方程解的存在性、有界性和集中性；2)综合运用Morse理论和极小极大原理研究具有次线性或渐近线性增长项的广义拟线性薛定谔方程解的存在性和多重性；3)结合变分法和上下解方法建立多重临界点定理，研究带扰动项和奇异项的广义拟线性薛定谔方程解的存在性和多重性等。本项目的研究将丰富和发展变分法与拟线性薛定谔方程理论，同时将促进微分方程与动力系统等相关领域的发展。

在本项目的开展过程中，我们主要深入研究了广义拟线性薛定谔方程等椭圆方程解的存在性及性态，具体研究内容包括：1）利用变量变换方法证明了一类广义拟线性Schrödinger方程当非线性项是超线性增长函数与次线性增长函数之和时存在非平凡解；2）利用Ekeland变分原理和山路引理证明了一类非齐次Kirchhoff-Schrödinger-Poisson系统正解的存在性和多重性；3）利用Nehari流形方法和集中紧性原理证明了一类拟线性椭圆方程基态解的存在性和集中性；4）运用临界点理论证明了一类Schrödinger-Poisson方程正则解的存在性、多重性以及无穷多径向对称解的存在性和收敛性；5）利用Nehari-Pankov流形方法证明了一类周期半线性椭圆系统基态解的存在和不存在性；6）利用极小极大原理证明了一类双调和方程满足临界指数增长时基态解的存在性；7）利用变分法证明了一类拟线性Choquard方程当非线性项满足双临界指数增长时解的存在性。本项目的研究丰富和发展了变分法与椭圆方程理论，同时促进了微分方程与动力系统等相关领域的发展。