Integral averages and oscillation criteria for half-linear partial differential equation.
The aim of this paper is to extend the technique of weighted integral averages to the half-linear partial differential equation \[ \Delta_p u+ c(x)\Phi(u)= 0,\tag{E} \] where \(\Delta_p u\equiv \text{div}(\| u\|^{p-2}\nabla u)\), \(p> 1\) is the \(p\)-Laplacian, \(\Phi(u)=| u|^{p-2} u=| u|^{p-1}\text{sgn\,}u\), and \(x= (x_i)^n_{i=1}\in \mathbb{R}^n\). On the one hand, the author obtains new oscillation criteria for (E) which can remove the disadvantage of some previous theorems, and on the other hand he shows that this technique allows to obtain oscillation criteria not only for the exterior of a ball, but also for different types of unbounded domains. Some illustrative examples are given.
scientific article; zbMATH DE number 2056329
scientific article