1967年，Trudinger提出了著名的Moser-Trudinger不等式，此后数学家们对该不等式做了深入的研究和推广，而且将Moser-Trudinger不等式广泛地用于偏微分方程和几何分析研究.本课题拟研究锥奇异Moser-Trudinger不等式.首先，在锥奇异空间上，我们计划得到有界域和无界域上的锥Moser-Trudinger不等式，进一步，我们拟研究锥Moser-Trudinger不等式的最佳嵌入常量问题和锥Moser-Trudinger不等式可达性问题.其次，我们拟研究高阶锥Laplace算子的性质，并利用该性质得到高阶锥Moser-Trudinger不等式. 最后，我们把锥Moser-Trudinger不等式用于偏微分方程研究，我们计划分别得到临界增长和次临界增长的一类方程的解的存在性和多重性.我还计划利用该不等式研究指数函数增长的非线性方程.

In 1967, the famous Moser-Trudinger inequalities were proposed by Trudinger. Since then, the Moser-Trudinger inequalities have been studied and popularized deeply and the Moser-Trudinger inequality is widely used in partial differential equation and geometric analysis by the mathematicians. This topic intends to study the Moser-Trudinger inequalities with conic singularities. Fistly, we plan to prove the conic Moser-Trudinger inequalities on bounded regions and unbounded regions in the spaces of conic singularities,respectively. Moreover, we will study the best embedding constant and the attainability of the conic Moser-Trudinger inequalities. Secondly, we will study the properties of the conic Laplacian of high order and use the properties to obtain the conic Moser-Trudinger inequalities of high order. Finally, we are going to apply the conic Moser-Trudinger inequalities to partial differential equations (PDEs). We need to obtain the existence and multiplicity results of solutions of a class of PDEs with critical growth and subcritical growth, respectively. We will also make a complete classification of solutions of a PDE with pure exponential growth.