非线性薛定谔方程半经典态的研究

项目名称： 非线性薛定谔方程半经典态的研究

项目编号： No.11201132

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 巴娜

作者单位： 湖北工业大学

项目金额： 22万元

中文摘要： 我们主要研究一类带有非负位势函数的非线性Schrodinger方程半经典态的存在性与衰减性等性质，其特点主要是：位势函数具有紧支集且在紧支集内部大于零，或者具有紧支集且在紧支集内部有零点。这会导致两个困难：在由这种位势函数诱导的赋范空间中得到的解未必属于L^2(R^N),故方程对应的能量泛函在H^1(R^N)上可能没有定义；方程对应的能量泛函一般也不满足PS条件。因此不能直接利用临界点理论来解决所提出的问题。我们将通过修正方程、构造合适的罚函数来克服这些困难，使得可以应用临界点理论得到修正方程的解。然后利用集中紧原理和先验估计等方法证明这个解就是原方程的半经典态，进而建立区域的几何形状特征、位势函数的性态与对应的稳态非线性Schrodinger方程半经典态之间的关系，同时证明解的衰减性等一些重要性质。对A.Ambrosetti在2007年提出的公开问题给出肯定地回答。

中文关键词： 非齐次半线性椭圆方程；非线性薛定谔方程；变分法；Jensen型二次泛函方程；稳定性

英文摘要： We will study the existence and decay properties of the multi-bump solutions to steady nonlinear Schrodinger equations with nonnegative potentials.The speciality is as follows:the potentials,which are always compactly supported, are either positive or have zeros in the interior of their supports. This leads to two troubles: one is that the solutions we hope to obtain in the normed space induced by potentials maybe not belong to the space L^2(R^N), which means that the energy functional derivated from original equation is not well-defined in the Sobolev space H^1(R^N); another is that it may not satisfy the PS condition. Consequently,direct application of critical point theory is forbidden to this functional. We will modify the equation and use penalization techniques such that the critical point theory can be directly applied to obtain the existence of the solution of the modified equation, then by the compactness concentation principle and apriori estimates we can prove this new solution is just the multi-bump solution of the original equation, and deduce the relations between the geometric characteristics of the domain,the behaviors of the potential functions and the multi-bump solutions of the corresponding steady nonlinear Schrodinger equations. Moreover,we also obtain some important properties such as decay

英文关键词： nonhomogeneous semilinear elliptic equation；nonlinear Schr？dinger equations；variational methods；Jensen-type quadratic functional equations；stability