On the asymptotic behavior of solutions of an equation of the Emden-Fowler type with a complex coefficient
Given the Emden-Fowler type of equation \[ y^{\prime\prime}(x)=p_{0}| y(x)|^{m}y(x), \] where \(x\in\mathbb{R}\), \(m>0\), and \(p_{0}\in\mathbb{C}-\mathbb{R}\). It is established that all non-continuable solutions are as follows: (1) a unique solution \(y\equiv0\) in \(\mathbb{R}\); (2) the solutions \(y=y(x)\) with \(| y(x)|=[mQ\cdot(x-x_{0})^{2}]^{-\frac{1}{m}}\), \(\text{arg}\,y(x)=-\frac{\text{Im}\,p_{0}}{(m+4)Q}\ln| x-x_{0}|+\varphi_{0}\) defined on \((-\infty,x_{0})\) or \((x_{0},+\infty)\) with arbitrary real \(x_0\), \(\varphi_0\), the sign \(+\) on \((x_{0},+\infty)\) and \(-\) on \((-\infty,x_{0})\), and the constant \(Q=\frac{m}{4(m+2)}(Re\,p_0+\sqrt{(\text{Re}\,p_0)^{2}+\frac{8(m+2)}{(m+4)^2}(\text{Im}\,p_0)^2})\); (3) the solutions defined on a finite interval \((x_1,x_2)\) with the asymptotic behavior \(| y(x)|=[mQ\cdot(x-x_{k})^{2}]^{-\frac{1}{m}}(1+o(1))\), \(\text{arg}\,y(x)=-\frac{\text{Im}\,p_{0}}{(m+4)Q}\ln| x-x_{k}|(1+o(1))\) as \(x\rightarrow x_k\), \(k=1,2\). The proof is based on an analysis of the dynamical system generated by the Emden-Fowler equation on the \(C^1\)-manifold \(\Phi\) obtained by endowing the quotient space of \(\mathbb{C}^2\setminus\{0\}\) given by the equivalence relation \((z_0,z_1)\sim(Az_0,A| A|^{\frac{m}{2}}z_1)\) for some \(A\neq0\) with an atlas consisting of two charts.
scientific article; zbMATH DE number 5319755