在科技和经济发展中, 很多重要的实际课题都需要求解偏微分方程, 为相应的工程设计提供必要的数据, 保证工程安全可靠且高效地完成任务. 非局部椭圆型方程或方程组在许多领域有着广泛的实际应用. 例如, 它常用于模拟反常扩散, 准地转流, 稳定的 Lévy 过程等物理现象. 从数学的角度来看, 非局部椭圆型方程主要来源于共形几何问题. 所以对于不同类型的非局部椭圆方程或方程组解的分类及相关性质研究是一个非常重要且有意义的课题. 由于这类问题本身的紧性缺失, 奇异性以及非局部性等特征, 因此对非局部方程或方程组的研究具有很大的挑战性. 本文主要通过直接移动平面法, 直接滑动的方法, 积分形式的移动球面法, 伸缩球面法和爆破分析技术分别对若干非局部椭圆方程和方程组的解的性质及其分类进行了研究. 主要结果有以下五个方面: 首先, 我们考虑非局部 tempered 方程解的性质. 针对该问题我们首先建立关于tempered 分数阶拉普拉斯算子的若干极值原理, 包括狭窄区域的极值原理以及无界区域的极值原理等. 然后结合直接移动平面法和直接滑动的方法得到了非局部tempered 方程解的对称性, 单调性和唯一性结果. 其次, 我们考虑一类带有耦合非线性项的混合阶共形不变方程组. 通过格林表示公式得到解的积分表达式；利用积分估计得到解在无穷远处的渐近行为, 最后结合积分形式的移动球面法得到了上述混合阶共形不变方程组解的分类结果. 再次, 我们考虑带有 Hartree 型非线性项的共形不变方程组. 过格林表示法, 积分形式的移动球面法和迭代的技巧得到了上述方程组非负解的完整分类结果. 从次, 我们考虑带有一般非线性项的混合阶椭圆型方程组. 首先通过积分估计以及格林表示法得到与该方程组等价的积分方程组, 进一步利用移动球面法得到该方程组解的分类结果. 最后, 我们考虑具有 Navier 边值条件的外部区域上的高阶 Lane-Emden 方程组.利用“Double Lemma”的方法得到上述方程组解的超调和性质, 从而得到了与原方程组相等价的积分方程组, 再结合伸缩球面法得到上述方程组的解的分类结果.
In the development of science, technology and economy, many important practical topics need to solve partial differential equations, to provide necessary data for the corresponding engineering design, to ensure that the engineering can complete the task safely, reliably, and efficiently. Nonlocal elliptic equations have attracted much attention due to its practical applications in other fields. For example, it has been frequently used to model diverse physical phenomena, such as anomalous diffusion, quasi-geostrophic flows, stable Lévy process, and so on. From the viewpoint of mathematics, nonlocal elliptic equations have been extensive interest in conformal geometry problems. Therefore, it is very important and meaningful to study non-local elliptic equations or system. Indeed, with loss of the compactness, singularity, and non-local properties, the study of the non-local problem will bring us new challenges. In this thesis, we are concerned with the classification of solutions to nonlocal elliptic equations (system) by the direct method of moving planes, sliding methods, the method of moving spheres in integral form, blow-up, and scaling spheres. First, we study various properties of solutions to the tempered fractional equations involving operators $-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}}$. We first establish various maximum principle principles, and develop the direct moving planes and sliding methods to establish symmetry, monotonicity, and uniqueness results of solutions to nonlocal tempered equation. Second, we consider the mixed order conformally invariant system with coupled nonlinearity in $\mathbb{R}^{2}$. We first derived the equivalent integral representation formula. Then we discuss the exact asymptotic behavior of the solutions to the system as $|x|\rightarrow \infty$. At last, by using the method of moving spheres in integral form, we give the classification of the classical solutions to conformally invariant system. Third, we consider the mixed order conformally invariant system with Hartree type nonlinearity. Combining the integral representation formula, the method of moving spheres, and the iteration technique, we derive classification results for non-negative solutions to the above system. Furthermore, we are concerned with mixed order elliptic system in $\mathbb{R}^{2}$. We first derive the equivalence between the PDE system and the corresponding IEs system. And then applying the method of moving spheres in integral form combined with integral inequalities, under certain assumptions, we give a complete classification of the classical solutions to the above system in $\mathbb R^{2}$. Finally, we consider higher-order Lane–Emden system in exterior domains with Navier boundary conditions. By proving the superharmonic properties of the solutions, we establish the equivalence between systems of partial differential equations and integral equations, then the method of scaling sphere in integral form can be applied to prove the nonexistence of the solutions.
