On the asymptotic behavior at infinity of solutions to quasi-linear differential equations

Abstract

Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation $$y^{(n)}+\sum _{j=0}^{n-1}a_j(x)y^{(j)}+p(x)|y|^k \mathop {\rm sgn} y =0$$ with $ n\ge 1$, real (not necessarily natural) $k>1$, and continuous functions $p(x)$ and $a_j(x)$ defined in a neighborhood of $+\infty $. For this equation with positive potential $p(x)$ a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. \endgraf Sufficient conditions are obtained for existence of solution to this equation which is equivalent to a polynomial.