一类非线性Hartree方程(组)变分方法的研究

本项目利用变分方法、临界点理论及分析技巧研究一类非线性Hartree方程(组)解的存在性、不存在性与多解性。对于Hartree方程，我们首先利用移动平面法、上下解的方法、集中紧性原理证明临界Hartree方程解的不存在性、存在性与多解性；其次我们利用Morse迭代、山路引理证明赝相对论下Hartree方程基态解的存在性，并通过极大值原理，Hopf引理及精细的估算证明解的指数衰减性。对于Hartree方程组，我们首先利用Ekeland变分原理、Lusternik-Schnirelmann原理证明临界Hartree方程组解的存在性与多解性；其次我们利用Nehari流形,截断函数与山路引理方法证明Hartree方程组基态解的存在性，并利用强极大极小值原理证明正基态解的存在性；最后我们利用隐函数定理、Miranda定理、Nehari流形证明非线性耦合Hartree方程组解的存在性。

In this project, the author applies the variational methods and the critical point theory and analytical techniques to study the existence, nonexistence, multiplicity of solution for the nonlinear Hartree system. On the hand, we study the nonexistence, existence and multiplicity of solution for critical Hartree via moving plane method, sub-superresolution techniques and concentration compactness principle, we also get the solution for pseudo-relativistic Hartree equation by Moser iterative technique and Mountain-pass lemma and obtain exponential decay of ground state solution via Maximum principle, Hopf lemma and its precise estimation. On the other hand. Firstly we study the existence of ground state solutions and multiple solutions for coupled critical Hartree system via using Ekeland variational methods and the Lusternik-Schnirelmann category theory. Secondly, we get the existence of ground state solution for nonlinear Hartree system via Nehari manifold, cut-off function and the Mountain Pass Theorem and obtain the positive ground state solution via strong maximum principle. Lastly, we get the existence of ground state solutions for coupled Hartree system via implicit function theorem, Miranda theorem, Nehari manifold.