For the equation y (n) + |y| k sgn y = 0, k > 1, n = 3, 4, existence of oscillatory solutions y = (x ∗ − x) −α h(log(x ∗ − x)), α = n ⁄ k − 1 , x < x∗ , is proved, where x ∗ is an arbitrary point and h is a periodic non-constant function on R. The result on existence of such solutions with a positive periodic non-constant function h on R is formulated for the equation y (n) = |y| k sgn y, k > 1, n = 12, 13, 14.
Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation y (n) + ∑ n−1 j=0 aj (x)y (j) + p(x)|y| k sgn y = 0 with n > 1, real (not necessarily natural) k > 1, and continuous functions p(x) and aj (x) defined in a neighborhood of +∞. 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. Sufficient conditions are obtained for existence of solution to this equation which is equivalent to a polynomial.