临界椭圆型方程和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 L\‘{e}vy 过程, 行星运动等物理现象. 从数学的角度来看, 临界椭圆型方程(组)主要来源于共形几何问题. 对于不同类型的临界椭圆方程和方程组解的相关性质研究是一个非常重要且有意义的课题. 本文主要通过有限维约化方法和局部Pohozaev恒等式, 结合格林表示和爆破分析等方法研究若干临界椭圆方程和方程组的解的存在性及非退化性. 主要结果有以下五个方面: 首先, 我们考虑了二阶临界 H\‘{e}non 型方程的爆破解的非退化性及其应用. 运用局部 Pohozaev 恒等式, 通过反证法证 明了临界 H\‘{e}non 型方程爆破解的非退化性. 作为非退化性的应用, 结合有限维约化方法, 我们构造性地证明了临界 H\‘{e}non 型方程的新解的存在性. 其次, 我们考虑了分数阶临界 H\‘{e}non 型方程的爆破解的存在性. 首先通过有限维约化方法将偏微分方程求解转化 为有限维代数方程求解, 再通过选取适当参数, 使得有限维代数方程有解, 由此我们构造性地证明了分数阶临界 H\‘{e}non 型方程的无穷多爆破解的存在性. 再次, 我们考虑了二阶临界 H\‘{e}non 型方程组的爆破解的存在性. 首先通过有限维约化方法将方程组求解转化为有 限维代数方程求解, 再精确计算能量泛函, 寻找适当参数使得方程有解, 由此构造性地证明了二阶临界 H\‘{e}non 型方程组的无穷多爆破解的存在性. 从次, 我们考虑了高阶曲率方程的爆破解的非退化性及其应用. 先建立局部 Pohozaev 恒等式, 并对局 部 Pohozaev 恒等式中的高阶项进行细致的估计, 通过反证法证明了解的非退化性. 作为非退化性的应用, 我们构造性地证明了高阶曲率方程的新解的存在性. 最后, 我们考虑了分数阶曲率方程的爆破解的非退化性及其应用. 首先对解进行提升, 对提升后的解建立局部 Pohozaev 恒等式, 再经过细致地计算和反证法证明了解的非退化性. 作为非退化性的应用, 我们构造性地证明了分数阶曲率方程的新解的存在性.

Critical elliptic equations and systems have a wide range of practical applications in many fields. For instance, they are often used to simulate anomalous diffusion, quasi-geostrophic flows, stable Lévy processes, planetary motion, and other physical phenomena. From a mathematical perspective, critical elliptic equations (systems) primarily arise from problems in conformal geometry. the study of the relevant properties of solutions to different types of critical elliptic equations and systems is a highly important and meaningful subject. In this paper, we primarily investigate the existence and non-degeneracy of solutions for several critical elliptic equations and systems using finite-dimensional reduction methods, local Pohozaev identities, Green‘s representations, and blow-up analysis. Our main results can be summarized in the following five aspects: Firstly, we consider the non-degeneracy and applications of blow-up solutions for second-order critical Hénon type equations. By utilizing the local Pohozaev identity and employing a proof by contradiction, we establish the non-degeneracy of blow-up solutions for the critical Hénon type equation. As an application of non-degeneracy, combined with finite-dimensional reduction methods, we constructively prove the existence of new solutions for the critical Hénon type equation. Secondly, we investigate the existence of blow-up solutions for fractional-order critical Hénon type equations. Initially, we transform the partial differential equation into a finite-dimensional algebraic equation using finite-dimensional reduction methods. Then, by selecting suitable parameters to ensure the solvability of the finite-dimensional algebraic equation, we constructively prove the existence of infinitely many blow-up solutions for the fractional-order critical Hénon equation. Next, we study the existence of blow-up solutions for second-order critical Hénon type equation systems. Utilizing finite-dimensional reduction methods, we convert the system of equations into a finite-dimensional algebraic equation for solution. By accurately calculating the energy functional and finding appropriate parameters that yield solutions, we constructively prove the existence of infinitely many blow-up solutions for the second-order critical Hénon type equation system. Furthermore, we investigate the non-degeneracy and applications of blow-up solutions for high-order curvature equations. We first establish local Pohozaev identities, estimating the higher-order terms in the local Pohozaev identity and proving the non-degeneracy of solutions using a proof by contradiction. As an application of non-degeneracy, we constructively prove the existence of new solutions for high-order curvature equations. Finally, we study the non-degeneracy and applications of blow-up solutions for fractional-order curvature equations. We enhance the solutions and establish local Pohozaev identities for the enhanced solutions. Through meticulous calculations and proof by contradiction, we demonstrate the non-degeneracy of the solutions. As an application of non-degeneracy, we constructively prove the existence of new solutions for fractional-order curvature equations.