全空间上变分形式的非线性椭圆型方程的研究

This project aims to use the variational method, the critical point theory and analysis methods to study the following problems:1.Researches on the existence, multiplicity and the asymptotic behavior of positive solutions of the Schrödinger equation, the field equation, the Kirchhoff equation, the Schrödinger-Poisson equation under only some restrictions on the limits of the coefficients at infinity;2. Researches on the uniqueness and non-degeneracy of positive solution of the limit equations of the nonlinear system, and construct multi-peak solutions to the nonlinear system with non-constant coefficients with these results;3. Researches on the existence, multiplicity and asymptotic behavior of the positive solutions of the nonlinear system when the coefficients are not periodic, do not have a well or anti-well structure, and do not have a fast decay rate, or when the coefficients are greater than their limits at infinity at some points and the coefficients are smaller than their limits at infinity at some points.. It is hoped that through the study of the above problems, we can find and propose new methods or improve the existing theories and methods to promote the development of the nonlinear functional analysis. At the same time, we hope to obtain a number of meaningful results, so as to contribute to the theoretical study of nonlinear partial differential equations.

本项目拟采用变分方法、临界点理论和分析方法来研究以下几个问题：1、在仅对系数函数在无穷远处的极限提要求的情况下,薛定谔方程,场方程,Kirchhoff型方程,薛定谔-泊松方程等问题的正解的存在性与多重性以及渐近性的研究；2、 R^N上的非线性方程组的极限问题的正解的唯一性和非退化性的研究，并藉此建立系数为非常数的方程组的多峰解的存在性结果；3、当系数函数不具有周期性,不具有井状或者反井状的结构,且不具有足够快的衰减率时，或者当系数函数在某些点处大于其在无穷远处的极限值同时又在某些点处小于其在无穷远处的极限值时,R^N上的方程组的正解的存在性、多重性与渐近性的研究。. 希望通过对上述问题的研究，发现并提出新的方法或改进现有的理论、方法，从而促进非线性泛函分析的发展。同时，希望获得一批有意义的研究结果，从而对非线性方程的理论研究有所贡献。

20世纪以来，来源于量子力学的非线性方程和来源于波色-爱因斯坦凝聚的非线性方程组受到了国内外数学界的广泛关注。非线性方程（组）的背景使得对其解的存在性及性态研究有着非常重要的意义。本项目对非线性方程和非线性方程组的正解的存在性与多重性等性态进行研究。我们证明了薛定谔-场方程正解的存在性和多重性，Kirchhoff型方程正解的存在性和渐近性以及非线性方程组正解的存在性和多重性等问题。通过对这些问题的研究，我们在一定程度上既揭示了系数函数的性态对方程(组)的正解的存在性，基态解的存在性等性态的影响，又揭示了方程组的指标和系数对方程组的正解的存在性和唯一性等性态的影响，这为物理学、材料科学等学科提供了重要的理论依据。