本项目将对一类拟线性椭圆问题和一类带非局部项的非线性椭圆问题的多解存在性及其相关性态展开系统的研究；所研究的椭圆问题包括源于理论物理的拟线性Schrodinger方程和源于弹性力学的Kirchhoff型方程。我们将借助于适当的变量替换和L.Jeanjean建立的泛函框架，证明一类不带(AR)条件，且带临界增长的拟线性椭圆方程的正解和无穷多变号解的存在性；同时，我们还将利用传统的平移平面法和打靶法，并借助于计算机实验来讨论一类特殊的拟线性椭圆问题正解的唯一性和k-波节解的唯一性；对带非局部项的非线性椭圆问题，我们将在较弱的条件下证明其正解和变号解的存在性。我们的研究不仅涉及到偏微分方程基本理论、非线性分析等重要的数学理论分支，同时也涉及到理论物理、弹性力学等应用领域。因此，本项目的研究是十分有意义的。

In this project, we will study the existence of multiple solutions and some properties of the solutions for a class of quasilinear elliptic problems and a class of nonlinear elliptic problems with nonlocal terms, which include the general quasilinear Schrodinger equations from theoretical physics and general Kirchhoff type equations from the free vibration of elastic strings. We will try to obtain the existence of positive soliton solutions for general quasilinear Schrodinger equations with critical growth and without (AR) conditions. Some uniqueness profiles for the positive solutions and k-node solutions will be discussed. Moreover, we will also try to prove the existence of positive solutions and sign-changing solutions for general Kirchhoff type equations. Our research is very important since it involves not only the theories in partial differential equations and nonlinear analysis, but also some applications areas such as theoretical physics and elastic mechanics.