The paper "Radial Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbf{R}^n$" by Yi Li and Wei-Ming Ni explores the symmetry properties of positive solutions to the nonlinear elliptic equation \[ \Delta u + f(u) = 0 \text{ in } \mathbf{R}^n, \quad n \geq 2, \] where \( f(u) \) is a given function. The authors build upon the method of "moving plane" introduced by A.D. Alexandroff and further developed by Gidas, Ni, and Nirenberg. They establish that if \( f(0) = 0 \) and \( f'(0) < 0 \), then all positive solutions must be radially symmetric about some point in \(\mathbf{R}^n\) and satisfy \( u_r < 0 \) for \( r = |x| > 0 \). The main result, Theorem 1, does not impose any assumptions on the asymptotic behavior of the solution \( u \) at infinity, addressing a gap in previous work. The proof relies on the strong maximum principle and the Hopf boundary lemma, showing that the solution must be radially symmetric and strictly decreasing away from the origin. The paper also discusses extensions to quasilinear and fully nonlinear equations, providing conditions under which solutions are radially symmetric. Examples are provided to illustrate the limitations of certain methods and to demonstrate the necessity of the assumptions in the main theorem. The authors conclude with a detailed proof of their main theorem, demonstrating that the solution must be radially symmetric about some point in \(\mathbf{R}^n\) and strictly decreasing away from that point.The paper "Radial Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbf{R}^n$" by Yi Li and Wei-Ming Ni explores the symmetry properties of positive solutions to the nonlinear elliptic equation \[ \Delta u + f(u) = 0 \text{ in } \mathbf{R}^n, \quad n \geq 2, \] where \( f(u) \) is a given function. The authors build upon the method of "moving plane" introduced by A.D. Alexandroff and further developed by Gidas, Ni, and Nirenberg. They establish that if \( f(0) = 0 \) and \( f'(0) < 0 \), then all positive solutions must be radially symmetric about some point in \(\mathbf{R}^n\) and satisfy \( u_r < 0 \) for \( r = |x| > 0 \). The main result, Theorem 1, does not impose any assumptions on the asymptotic behavior of the solution \( u \) at infinity, addressing a gap in previous work. The proof relies on the strong maximum principle and the Hopf boundary lemma, showing that the solution must be radially symmetric and strictly decreasing away from the origin. The paper also discusses extensions to quasilinear and fully nonlinear equations, providing conditions under which solutions are radially symmetric. Examples are provided to illustrate the limitations of certain methods and to demonstrate the necessity of the assumptions in the main theorem. The authors conclude with a detailed proof of their main theorem, demonstrating that the solution must be radially symmetric about some point in \(\mathbf{R}^n\) and strictly decreasing away from that point.