Note on asymptotical behavior of solutions of Emden-Fowler equation and the existence and uniqueness of solution of some Cauchy problem

摘要

We study differential equation of Emden-Fowler type $$y'' - x^{a}y^{sigma}=0,quadmbox{for $ainmathbb{R}$ and $sigma0$.}$$ We describe the conditions on parameters $a$ and $sigma$ which assure that this equation has infinitely many solutions defined in some neighborhood of zero and the conditions which guarantee existence of infinitely solutions with certain asymptotic behavior.