两类具有梯度项和奇异性非线性微分不等式的研究

This project mainly studies the two classes of nonlinear elliptic differential inequalities with gradient nonlinearities and singularities in bounded and unbounded domain. This kind of problem has a broad prospect of application, and has become a hotspot in recent years' research of partial differential equations. We mainly study the nonexistence results of the solutions to the coercive and anticoercive nonlinear elliptic differential inequalities by using the test function method developed by Mitidieri and Pohozaev. The key to the study of nonlinear elliptic differential inequalities is a special choice of test functions associated with the nonlinear problem considered. By various computing skills and the estimation method we obtain an a priori estimate for a solution to a nonlinear problem considered. Then, the nonexistence of a solution is proved by contradiction. Note that this choice is determined by both nonlinear and singular characters of the problem and depends on the domains considered. On the base of many researches at home and abroad, we generalize previous studies and use the test function method and analysis techniques to overcome the difficulties, and discuss further the two classes of nonlinear elliptic differential inequalities.

本项目主要在有界区域和无界区域上研究两类既有梯度项又有奇性的非线性椭圆型微分不等式，该问题具有广泛的应用前景，已成为非线性偏微分方程研究中的热点问题之一。我们主要应用俄罗斯数学家 Mitidieri 和 Pohozaev 所创立的试验函数方法，研究强制和非强制非线性椭圆型微分不等式解的非存在性。研究这类问题的关键是如何构造出合适的试验函数，并结合各种技巧运算和估计方法得到解的先验估计，再应用反证法即可证明解的非存在性。需要注意不等式右端非线性项中带有梯度项会对建立所需的先验估计带来困难，并且区域和奇性的不同都会影响到试验函数的选取。在广泛总结前人工作的基础上，结合我们已有的研究成果，运用本方向各种技巧运算和估计方法克服这些困难，对这两类非线性椭圆型微分不等式进一步探讨。

非线性微分不等式解的存在与非存在性是偏微分方程理论与变分不等式理论中的重要问题，具有广泛的实际应用背景，如数学物理中的障碍问题、自由边界问题、偏微分方程中的上下解以及偏微分方程的解所满足的重要不等式等都属于这类问题。本项目主要对以下几个方面进行了研究：(1)研究一类各项奇异的拟线性微分不等式解的正性和非存在性。(2)考虑一类具有奇性的非线性常微分微分不等式(组)解的存在与非存在性。(3)研究一类具有梯度项和奇性的微分不等式(组)解的存在与非存在性。(4)讨论了具奇异边界条件的非线性椭圆型方程解的存在性与非存在性。