含临界指标的非线性椭圆问题的临界维现象

The nonlinear elliptic problems including the critical exponents posses their profound background in theoretical physics and geometry. It is very complicate when people try to deal with the nonlinear elliptic problems including the critical exponents since the critical term appear in the nonlinear part which cause that the existence and nonexistence of positive solution and sign-changing solution depend on the dimensions of the domain, topology structure of the domain, the homogeneous perturbation or the nonhomogeneous perturbation. The main purpose of this project is to explore the existence and nonexistence of solutions affected by the dimensions of the domain for some nonlinear elliptic problems including the critical exponents which will cover qusilinear Schodinger equations, biharmonic equations, multiple harmonic equations, p-biharmonic equations and the equations with the critical Hardy terms. Because the critical term and critical Hardy term cause the loss of the compactness for the corresponding variational functional, the standard variational methods can't be applied directly and thus people have to develop some new functional methods and analysis technique to investigate the phenomenon caused by the critical exponents and Hardy term. Therefore, our project is meaningful both in pure mathematics and applying mathematics.

含临界指标的非线性椭圆问题有它深刻的物理背景和几何背景。由于问题中非线性项含临界指标使问题的研究变得相当复杂。空间的维数、区域的拓扑结构、齐次或非齐次扰动都会导出很多非常有趣的存在性与非存在性结果。本项目的主要目的是探讨空间的维数对几类非线性椭圆问题解的存在性与非存在性以及多解性的影响。其研究结果将覆盖拟线性Schodinger方程、双调和方程、多重调和方程、p-双调和方程以及含临界Hardy项的各类方程。由于方程中出现临界指标或临界Hardy项从而导致对应的变分泛函失去紧性，标准的变分方法不能直接应有，因此很多奇怪的现象由此产生，需要人们进一步发展泛函工具和分析技巧来探讨这些奇怪的现象。因此，本项目的研究涉及到非线性分析、几何拓扑等数学理论分支，同时也涉及到理论物理、反应扩散等重要的应用领域。

对一类广义拟线性Schrodinger方程，我们发现了它所对应的临界指标，并对这类带临界指标的拟线性椭圆问题，我们得到了其驻波解的存在性；对带非局部项的Kirchhoff-type问题和Schrodinger-Poission 方程，我们定义了其变号基态解，并在一定条件下证明了其变号基态解的存在性，同时，我们还研究了当问题中的非局部项消失时，对应的变号基态解的渐近行为；对源于微分几何的数曲率方程，我们证明了其峰值解的局部唯一性，这结果的一个直接推论是：如果曲率函数K(y)关于某一个变量是周期的，则对应的数曲率方程存在无穷多个关于该变量是周期的解；对非齐次Bose–Einstein Condensation (BEC) 问题和Hartree问题，我们得到了其对应的能量泛函在某种意义下约束极小的存在性与非存在性；对带零位势或Hardy位势的椭圆问题，对含临界增长的椭圆方程组我们得到了一些有趣的存在性与非存在性结果。