带临界Hardy-Littlewood-Sobolev（简称HLS）指数的积分方程与积分曲率问题、Minkowski问题等几何问题相关。关于其解的存在性的研究有异于带临界Sobolev指数的椭圆方程的本质困难：有界PS序列没有逐点收敛性；负指数情形紧嵌入不等式和集中紧性原理不成立。项目围绕两种思路研究有界区域上这类方程的正解存在性和多重性：对区域施加几何或拓扑条件；对基本方程施加扰动项。针对正指数情形拟使用紧嵌入不等式和全局爆破分析克服PS序列没有逐点收敛性的困难，结合集中紧性原理证明全局紧性结果；针对负指数情形拟使用全局爆破分析克服紧嵌入不等式不成立的困难，再结合反向HLS不等式和能量估计等获得局部紧性条件。最后使用变分法和临界点理论证明正解存在性和多重性。项目将建立有界区域上带临界HLS指数的积分方程正解存在性的系统理论，明确其与椭圆方程的本质区别，为相关几何问题的解决提供理论支撑。

The integral equations involving critical Hardy-Littlewood-Sobolev（HLS in short）exponents have close relationship with some problems arising from geometry, such as the integral curvature problem and the Minkowski problem. There are the following essential difficulties different from that arising from the elliptic equations involving critical Sobolev exponents in studying the existence of solutions to this kind of integral equations: the bounded PS sequences may not converge pointwise; for the integral equations involving negative exponents, the compact embedding inequality and the concentration compactness principle even do not hold anymore. The project devotes to the existence and multiplicity of positive solutions to the integral equations involving critical HLS exponents on bounded domains along two lines：assuming geometric or topological conditions on the domains; adding perturbing terms to the fundamental integral equations. For the integral equations involving positive exponents, we will first use the compact embedding inequality and the global blowup analysis to overcome the difficulty that the bounded PS sequences may not converge pointwise, and then combine with the concentration compactness principle to prove the global compactness results; for the integral equations involving negative exponents, we will use the global blowup analysis to overcome the difficulty that the compact embedding inequality does not hold, and then combine with the reversed HLS inequality and the energy estimates to obtain the local compactness conditions. At last the variational methods and critical point theory will be used to prove the existence and multiplicity of positive solutions. The project aims to establish the systematic theory of the existence of positive solutions to the integral equations involving the critical HLS exponents on bounded domains, to clarify the differences between the integral equations and the elliptic equations, and to provide theoretical support to the corresponding problems in geometry.