静态非局部薛定谔方程的奇异位势理论和定性分析

The study of static nonlocal Schrodinger equations involving singular potentials is a focus of attention now because they are widely applied in quantum mechanics, N-body dynamics, physics of laser beams , condensed matter physics and other fields. In addition, the study of such a class of equations also promotes the progress of other mathematical branches such as the nonlinear analysis, the classical inequalities and the geometric analysis. This proposal is concerned about several static nonlocal Schrodinger equations involving singular potentials, including the Choquard type equations, the Hartree type equations, and the Maxwell-Schrodinger equations, as well as the Lane-Emden equations with the minus exponents appearing in geomtry, the Euler-Lagrange equations related to the sharp functions of the Hardy-Littlewood-Sobolev inequality. We are going to study the qualitative properties such as existence, integrability, regularity, decay estimate and classification of positive solutions. Such a qualitative analysis can help us to well understand the shapes of solutions. Those qualitative properties will be verified by converting those equations to the corresponding integral equations involving the Riesz potentials, the Bessel potentials or the Wolff potentials.

含有奇异位势的非局部薛定谔方程广泛出现在量子力学，多体动力学，激光束物理，凝聚态物理等学科中，如今已受到越来越多的关注。在数学领域内，此类方程的研究也促进了非线性分析，经典不等式和几何分析的发展。 本课题旨在研究几类静态非局部薛定谔方程, 包括Choquard型方程, Hartree型方程, Maxwell-Schrodinger方程,以及其它相关方程，包括来源于几何中的带有负指数的Lane-Emden型方程，Hardy-Littlewood-Sobolev不等式最佳函数满足的Euler-Lagrange方程。我们将讨论正解的定性性质，包括存在性，可积性，正则性，衰减估计和正解的分类等。 这些定性性质能帮助我们更好地刻画解的形状。通过把方程转化为含有 Riesz位势，Bessel位势或Wolff位势的积分方程，并对这些积分方程进行位势积分估计，我们希望得到上述非线性方程的定性性质。

含有奇异位势的非局部薛定谔方程具有鲜明的物理背景，并且与多个当今数学领域的热点问题的研究密切相关。本课题研究了几类静态非局部薛定谔方程, 包括Choquard型方程, Hartree型方程, Maxwell-Schrodinger方程, 带有负指数的Lane-Emden型方程，Hardy-Littlewood-Sobolev型积分方程。我们探讨了正解的定性性质，包括存在性与几类临界条件的关系，压缩映射诱导的正则定理提升的可积性，衰减估计和正解的分类等。几个重要的椭圆方法起到关键作用，包含移动平面法，正则提升引理，双倍引理，以及Moser迭代等。