本项目将应用变分方法和临界点理论研究几类非线性变分问题的多解性，以及解的拓扑、几何和分析性态等。研究内容包括：(1) 拟线性椭圆变分问题，此类问题可以用来刻画非牛顿流体、非线性弹性问题以及孤立波的传播现象等，我们将利用极小极大定理、Morse理论等变分方法研究非平凡解的存在性和多解性；(2) Schrodinger-Poisson方程，此类方程在量子力学、半导体理论等领域有着广泛的应用，我们将研究非线性项带有凹凸项时多个正解及无穷多解的存在性，以及研究不同的位势函数对方程解的存在与几何性质的影响等。本项目所选课题是我们在近年来研究中所遇到的新问题，具有重要的理论意义和研究价值。我们期望通过本课题的研究，推进非线性分析理论与应用的发展。

In this project we will study the existence and multiplicity of solutions for several nonlinear variational problems by the variational methods and critical point theory. First of all，using the minimax theory and Morse theory we want to get the existence and the topology theories of multiple nontrivial solutions for a class of the quasilinear elliptic equations, which arise naturally in various contexts of physics, for instance, in the study of propagation phenomena of solitary waves, non-Newtonian fluids and nonlinear elasticity problems. Secondly, the Schrodinger-Poisson equations will be considered, which come form the quantum mechanics，the theory of semiconductor and so on. Using the variational methods we will study the multiple solutions of this system. For example, we assume that the nonlinearity of this problem may involve a combination of concave and convex terms,and we want to obtain infinitely many solutions. The topics of this project are included in the international frontiers of the scientific research of variational theory, and these problems have important theoretic meanings and research values. We hope to make some contributions to the development of nonlinear analysis via the study of this project.