拟线性薛定谔方程及其相关问题的研究

In recent years, the study on the quasilinear Schrödinger equation is the hot research field in nonlinear analysis. The modified Schrödinger equation and its generalized quasilinear Schrödinger equation are two important classes of quasilinear Schrödinger equations. The project is to study them by using the variational and topological methods. It generalizes a large number of results on Schrödinger equations. The contents are organized as follows: studying the existence of multiple solutions in the critical case and the existence of the high energy solution in the subcritical case for the modified Schrödinger equation, respectively. In order to obtain existence theorems under weaker conditions, we must overcome the impact of the modified term on the energy functional (e.g. the geometry structure of the functional, the boundedness and compactness of the Palais-Smale sequences and so on). In addition, we are concerned with the existence of infinitely many solutions under the periodic or non-periodic conditions for the generalized quasilinear Schrödinger equation. For the related singularly perturbed problems, we focus on the existence and asymptotic behaviors of their solutions by variational methods. The completion of these problems not only reveals the new laws, but also has an important academic value and broad application prospect.

近年来，拟线性薛定谔方程是非线性分析中的热点研究领域。修正薛定谔方程和广义拟线性薛定谔方程是两类重要的拟线性薛定谔方程。本项目拟利用变分方法和拓扑方法对其进行研究，这推广了大量的薛定谔方程的结果。具体来说：1.研究修正薛定谔方程临界情形下多解的存在性和次临界情形下高能量解的存在性。主要克服的困难是修正项对能量泛函的影响（如：泛函的几何结构，Palais-Smale序列的有界性和紧性等）。在更弱的非线性条件下建立相应的存在性定理。2.研究广义拟线性薛定谔方程无穷多解的存在性以及相应奇异扰动问题解的存在性与集中性。对于无穷多解的存在性，我们将分别在周期条件或非周期条件下展开研究。对于奇异扰动问题，利用变分方法并适当加以改进来研究解的存在性与渐近性态。这些问题的解决不仅能揭示出新的规律，而且具有重要的学术价值和广泛的应用前景。

具有二阶导数项的拟线性薛定谔和薛定谔泊松方程是许多物理现象的模型。例如：等离子物理，流体力学，耗散系统量子力学等等。我们利用变分方法来证明这两类方程的非平凡解的存在性。. 对于拟线性薛定谔方程，当其非线性项分别是渐近立方和超立方条件下，我们证明了高能量解的存在性。当位势函数具有临界频率时，我们证明了局部解的存在性以及解集中在位势函数的局部最小值的某个孤立分支中。利用Nehai流形方法和LS稠数理论，我们证明了临界指标情形下多解的存在性。.我们利用广义Nehari流形方法得到了薛定谔泊松系统高能量解的存在性。.对于具有临界指数的拟线性薛定谔方程组，我们证明了多重解的存在性。