Uniqueness of solutions to second order Emden-Fowler type equations with general power-law nonlinearity (Q2027468)

In the present article there is considered the following second order differential equation \[ y''=p(x,y,y')|y|^{k_0}_{\pm}|y'|^{k_1}_{\pm}, \quad k_0>0,\quad k_1>0\tag{1} \] where \(|a|^b_{\pm}\) denotes \(|a|^b \operatorname{sgn} a\), \(p\) is a positive continuous function and is locally Lipschitz in the last two parameters. It is also assumed that the equation (1) satisfies the initial condition \[ y(0)=y_0, \quad y'(0)=y_1.\tag{2} \] The main results are obtained in the second chapter. The author obtains the conditions on the constants \(k_0\) and \(k_1\) (\( k_0\geq1, k_1\geq1\); \(k_0<1, k_1\geq1\); \(k_0\geq1, k_1<1\); \(k_0<1, k_1<1, k_0+ k_1\geq1\) and \(k_0+ k_1<1\)) and on the initial values providing uniqueness or nonuniqueness of solutions to the initial value problem (1)--(2).