We study oscillatory properties of the second order half-linear difference equation \[ \Delta (r_k|\Delta y_k|^{\alpha -2}\Delta y_k)-p_k|y_{k+1}|^{\alpha -2}y_{k+1}=0, \quad \alpha >1. \qquad \mathrm{(HL)}\] It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation \[ \Delta (r_k\Delta y_k)-p_ky_{k+1}=0. \] We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.
In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm H in ℝ n which is continuously differentiable for x ≠ 0 and such that Hp is strictly convex for some p > 1. Two important special cases are the p-Laplacian and the so-called pseudo p-Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria for the nonexistence of positive solutions in exterior domains for such equations by means of comparison with the equation exhibiting a kind of ''anisotropic radial symmetry''.