Growth conditions for oscillation of nonlinear differential equations with p-Laplacian

In this paper, oscillation criteria are established for all solutions of second-order nonlinear differential equations of the form (ϕp(x′))′+1tpg(x)=0,t>0. Here ϕp(y) is the one-dimensional p-Laplacian operator, and g(x) satisfies the signum condition xg(x)>0 if x≠0 but is not assumed to be monotone. The equation naturally includes the famous Euler differential equation and half-linear differential equations. The main purpose is to examine the influence of certain growth conditions of the nonlinear term g(x) on the oscillation of solutions. The conditions are shown to be sharp. Some of differential inequalities play important roles to prove our results. A simple example is included to illustrate the main result. A conjecture on the inverse problem is also given. Finally, elliptic equations with p-Laplacian operator are discussed as an application to our results.