两类非线性椭圆方程解的研究

By using the critical point theory and the analysis technologies, we try to study the following problems: (1) Suppose that the nonlinearity satisfies B-L conditions. By seeking for new potential conditions, we prove that Schrodinger equation and Kirchhoff equation have ground state solution and bound state solution. (2) By seeking for the conditions of the best critical growth, we prove that autonomous Schrodinger equation and Kirchhoff equation have ground state solution and bound state solution. (3) Suppose that the nonlinearity satisfies the conditions of the best critical growth. Under the new potential conditions, we prove that Schrodinger equation and Kirchhoff equation have ground state solution and bound state solution.

我们拟利用临界点理论和分析技巧来研究下列问题: (1) 探索新的位势条件, 使得在非线性项满足B-L条件之下, 薛定谔方程和基尔霍夫方程具有基态解、束缚态解. (2) 探索最佳的临界增长条件, 使得自治的薛定谔方程和基尔霍夫方程具有基态解. (3) 在新的位势条件和最佳的临界增长条件之下, 研究薛定谔方程和基尔霍夫方程的基态解、束缚态解.

本项目研究了极具物理背景的Schrödinger方程、Kirchhoff方程等数学模型，在恰当的位势条件和非线性增长条件下，利用变分方法和分析技巧得出了非平凡解的存在性、多重性（主要包括正解、基态解、束缚态解等）及解的相关性态，发表了学术论文14篇（其中11篇论文被SCI期刊收录），组织召开了国内学术会议1次，参加国际、国内学术会议40余人次。