几类物理、几何和生物模型中正解的存在性

The program describes some models from Physics, Geometry and Biology. In these models, positive solutions play the key role since in the most cases, the unknown function represents density(mass). So the key problem is to study the existence of positive solutions. First, we consider Schrodinger equation and Schroding system where the potentials go to some positive constants with slow decay. We want to build the multiplicity of positive solutions. Because of the slow decaying rate, the typical Sobolev frame is invalid. We need to define new frame to match the gap between the potential and the convergent constant. On the other hand, for the system when the dimension of space gets large, the interaction terms become sublinear or just linear, such phenomena can also be observed in Bose-Einstein system, which brings us the difficulty in the construction of concentration. Hence we hope to find a new technique to overcome the linearity. Next we will consider some kinds of equations involving the partial derivative of time with fractional Laplacian operator. Since the classical Laplacian operator gives us so rich solutions, we naturally try best to make the fractional Laplacian operator clear in the effects to the existence, like concentration, non-trivial in infinity, transition layers, and so on. Finally, we focus on biharmonic equation with weighted function in conformal matrices. Due to the weighted function, non-radial solutions will appear in the symmetric domains. Also in the bounded domain, the weighted function will enlarge the interval of existence. There is a relationship between the weighted biharmonic equation and the weighted Laplacian system, thus we would like to study the system and then apply the results to biharmonic equation. The above problems are very interesting and meaningful. Through this program, one hand, we expect to deal with these problems very well. On the other hand, we also hope to build new theory and techniques which will be used widely in other models and manifolds.

本项目描述了几类物理、几何和生物中的模型，着重探究这几类方程(组)正解的存在性。首先是来自于量子力学的薛定谔方程和方程组，研究电势能函数在无穷远处以较慢的速度收敛于一正常数时，论证无穷多正解的存在性。在空间维数比较大时，薛定谔方程组中两种粒子的交互作用项呈现出(次)线性，该现象也体现在玻色-爱因斯坦方程组中。本项目希望摸索出一种新的构造技巧，能够处理次线性项，从而得到多个正解的存在性。其次，相对于拉普拉斯算子描述的布朗运动和随机扩散，同时拉普拉斯算子折射出了丰富的解集，数学家们近几十年来积极关注分数阶拉普拉斯算子，探究该算子的作用机理。本项目考察具有分数阶拉普拉斯算子的椭圆方程和抛物方程各类正解的存在性，例如凝聚解，无穷远处不趋于零的正解，相变层交替出现而且具有多个端点的解。最后考察一类共形几何中的双调和方程，探讨该方程中的权函数对解集的作用机制，进而刻画出临界态时带权方程组的解集。

对椭圆方程(组)和反应—扩散方程(组)的凝聚及爆破现象的研究是最近三十年偏微分方程研究的一个热点。这些研究丰富了我们对方程解的理解。本项目一方面旨在通过研究分析，挖掘出物理、几何和生物中更多有意义的凝聚现象；另一方面，探讨几何学中的预定曲率及预定曲率的奇异解以及正则解分析。这类分析很大程度上依赖于对爆破(blow up)解的研究，因此带参数的对称解和奇异解(blow-up solutions)极其重要。故而我们首先研究了径向对称解，再以此为基础，探讨奇异解的存在性，进而论证其它解的存在性，从而刻画出方程产生丰富解集的本质原因。 . 本项目主要考察了几类物理、几何和生物数学中的模型，研究了这些模型解的存在性(concentration)、解的分析性质和渐近行为。重要成果体现在：标注本项目号的SCI论文共计发表17篇。这些论文主要表现在:.(1) 在几类物理和几何模型中呈现了丰富的凝聚现象；.(2) 对几类四阶椭圆型方程得到了刘维尔型结果和正则解/奇异解的刻画；.(3) 研究了几类生物种群模型，在不同的参数假设下，我们获得了各类渐近性态，取得了一系列的研究成果。.这些成果充分展示了偏微分方程解的多样性、复杂性和生动性，获得了同行的一致肯定，为该领域的发展提供了一定的学术价值。