Growth conditions for oscillation of nonlinear differential equations with \(p\)-Laplacian
The authors give criteria for the oscillation of all solutions of the nonlinear differential equation \[ \left( \phi_p(x')\right)'+\frac{1}{t^p}g(x)=0, \;t>0, \tag{1} \] where \(\phi_p(y)\) is a real-valued function defined by \(\phi_p(y)=| y| ^{p-2}y\) with \(p>1\) a fixed real number, and \(g(x)\) is a continuous function on \(\mathbb{R}\) satisfying \(xg(x)>0\), if \(x\neq0\), and a suitable smoothness condition for the uniqueness of solutions of the initial value problem. The equation includes the famous Euler differential equation and half-linear differential equations. The main result of the paper is the following theorem: Assume \(xg(x)>0\) for \(x\neq0\) and suppose that \[ \frac{g(x)}{\phi_p(x)}\geq \left( \frac{p-1}{p} \right)^p+\frac{\lambda}{(\log| x| )^2} \] for \(| x| \) sufficiently large, where \[ \lambda>\frac12 \left(\frac{p-1}{p} \right)^{p+1}. \] Then, all nontrivial solutions of (1) are oscillatory. The authors conjecture a sufficient condition for all nontrivial solutions of (1) to be nonoscillatory and apply their results to elliptic equations.
scientific article; zbMATH DE number 2173083