带有衰减位势的非线性薛定谔方程的解

The variational method is one of the main theories in nonlinear functional analysis and an important research area in modern mathematicas as well, with very extensive and deep applications in areas such as nonlinear differential equations. However, the energy functional with respect to the differential eqiuation is not well defined in classical Sobolev space when the potential is tending to zeo at infinity. And so the variational method could not show its powerfulness. Moreover, to study the rich properties of the solutions of some special differential equations will help us to understand general differential equations in deep. The content to study include mainly: the asymptotic behavior and radial symmetry of positive solutions of semilinear equations with Hardy potential in the annulus; the nodal solutions of nonlinear Schrodinger equations with potentials tending to zero at infinity. Those problems listed above are not only highly important mathematical problems, being located at the forefront of the research area of nonlinear analysis at the international lever, but also problems with great diffficulties, new methods and ideas being needed in order to solve them. In this project we will use variational method and topological methods combined with various techniques in analysis to study several important problems in nonlinear elliptic equations, looking forword to obtain good results.

变分方法是非线性泛函分析的主要工具之一，是现代数学的重要研究领域，在非线性微分方程等领域有非常广泛和深刻的应用。但是，当微分方程具有衰减位势时，微分方程所对应的能量泛函在通常的Sobolev空间没有定义，这时，变分方法便不能施展它的威力。另外，研究一些特殊方程解的性质，往往能帮助我们理解更一般的微分方程。本项目拟研究的问题就是这些情形，具体内容如下：在环域上带有Hardy项的超临界椭圆方程的唯一径向对称正解的渐近行为；带有衰减位势的非线性薛定谔方程的变号解的存在性。这些问题既是重要的数学问题，处于国际非线性分析领域的前沿，同时也是非常困难的问题，解决起来需要新的方法和思路。本项目将结合变分方法，拓扑方法，截断技巧和爆破技巧，对上述两类偏微分方程进行研究，期待能做出出色的工作。

本项目应用变分方法并结合拓扑方法，以及各种分析工具特别是截断技巧和爆破技巧研究下面两类非线性椭圆型偏微分方程。一类是带有衰减位势的非线性Schrodinger方程，在位势具有局部山路结构的条件下，我们证明了非线性Schrodinger方程存在多个不同的变号解。并且，通过对解做爆破分析,我们得到这些解具有集中现象；通过构造比较函数，我们还得到了这些解在无穷远处的衰减性估计。研究结果表明: 位势函数在无穷远处的衰减速度及非线性项在无穷远处的增长速度将影响解在无穷远处的衰减阶。我们研究的另外一类方程是在环域上带有Hardy项的超临界椭圆方程。本项目主要研究方程的径向对称正解的渐近行为和非退化性质。我们得到此微分方程的径向对称正解在p趋近于正无穷时满足的极限方程和解析表达式。同时也得到了Soboleve-Poincare嵌入常数的极限和解的最大值的渐近展开式。另一方面，由于微分方程不是自治的，我们通过Flower变换将方程转化为一个自治方程，通过精细的分析，得到解的非退化性质。