对非线性变分问题的研究是国际数学界、非线性分析领域中一个非常活跃的研究方向。本项目的主要目的是对几类有很强应用背景的非线性椭圆型方程及方程组解的存在性，渐近行为等定性性质进行研究。具体来讲，我们首先利用变分法研究几类非线性薛定鄂方程及方程组，讨论其正解，变号解以及多解的存在性、渐近行为等； 然后研究Kirchhoff 类型的偏微分方程， 讨论其基态解，多解的存在性及渐近行为。最后我们研究几类分数阶偏微分方程，通过讨论其对应的局部问题，我们利用变分法研究其基态解，多峰解存在性及渐近行为。对这些问题的研究，不仅涉及到非线性分析，而且也涉及到拓扑等理论分支。因此，我们的研究不仅可以回答很多应用问题，也可以推动一些数学理论分支的发展。

Study on nonlinear variational problems is a hot field in nonlinear analysis arround the world. Our main task of this project is to study a class of nonlinear elliptic partial differential equations and systems which have strong applied background. More precisely, first, we study a class of nonlinear Schrödinger equations and systems, we study the existence of positive solution, sign changing solution and also the multiplicity of solutions, we also study the asymptotic behavior of these solutions. Then we study Kirchhoff type equations, we study the existence of ground state solution, bump solutions and also the asymptotic behaviors of these solutions. In the end, we study a class of fractional partial differential equations. By considering the corresponding local problem, using variational methods, we study the existence of ground state solution, bump solution and also the asymptotic behavior of these solutions. Our research to these problems can not only solve some applied problems, but also it is important to the development of mathematical theory.