一类带有两种非局部项的非线性椭圆方程中若干问题研究

In this project we will study the existence of solutions and the properties of the solutions for a class of nonlinear Choquard equations perturbed by the Kirchhoff type nonlocal term via variational methods and critical point theory. Arised from quantum physics，this class of Schrodinger equations have many important applications in nonlinear optics, electromangetics, condensed matter physics, etc, and become a hot spot of current research in the field of nonlinear functional analysis. The equations have attracted the interest of many mathematical researchers, and obtained a large number of outstanding research achievements, because of the lack of compact condition and the special structure of the equations, many challenging research topics are left. This project contains the following two aspects:.(1) we will study the existence of positive solutions, sign-changing solutions of the equation under the interaction between the two nonlocal terms, forthermore, we will also investigate the asymptotic behaviors of the solutions;.(2) we will study the optimal ranges of parameters such that normalized solutions of the equation exist, and the concentration phenomenon of the normalized solutions will also be considered.

本项目拟利用变分方法及临界点理论研究一类带有Kirchhoff 型非局部项的Choquard 方程解的存在性及解的性态问题。这类方程起源于量子物理，它在非线性光学、电磁学、凝聚态物理等领域中有着许多重要的应用，是当今非线性分析领域的研究热点，吸引了众多数学研究者的兴趣，涌现出了大量突出的研究成果，同时由于缺乏紧性条件以及方程本身特殊的具体结构，留下了一些具有挑战性的课题。本项目的研究包括以下两个方面：.(1) 研究该方程中两个非局部项之间的相互作用对方程正解、变号解存在性的影响，并探究解的渐近性态；.(2) 研究该方程中两个非局部项之间的相互作用下使得方程正规化解的存在的参数的最佳存在范围，并研究正规化解的集中现象。