Existence and uniqueness of solutions of some Cauchy problems for the Emden-Fowler equation (Q1982379)

The article is devoted to the following the nonlinear Emden-Fowler differential equation \[ y''-x^a y^\sigma=0,\tag{1} \] where \(a, \sigma\) are real parameters and \(\sigma \neq1\). The article consists of 3 sections and the list of references. The first section is devoted to the general introduction into the problem. In Section 2, the authors consider the equation (1) with the following initial conditions \[ y(0)=c, \quad y'(0)=\lambda , \quad c\in(0;+\infty), \lambda \in \mathbb{R}\cup\{-\infty\}. \tag{2} \] There are obtained necessary and sufficient conditions on the parameters \(a, \sigma\) of the equation (1) for the Cauchy problem (1), (2) to have (or not to have) a solution and about uniqueness of such solutions in case its existence. An example shows that the solution of the Cauchy problem (1), (2) is not necessarily unique in case \(\lambda=-\infty\). Section 3 of the article is devoted to the equation (1) with the following initial conditions \[ y(x_0)=c, \quad y'(x_0)=\lambda , \quad x_0>0, \quad a, \lambda \in \mathbb{R}\cup\{0\}, \sigma <0.\tag{3} \] Theorem 3.1 gives the existence of a solution of the Cauchy problem (1.1), (1.3). The question arises about the uniqueness of such a solution. For \(\lambda > 0 (\lambda < 0)\), the answer was obtained in a Theorem 3.2., the case \(\lambda = 0\) was considered in the theorem 3.3. The results presented in the article are very important for the further study of solutions of Emden-Fowler differential equation.