Influence of nonlinear perturbed terms on the oscillation of elliptic equations

Our concern is to solve the nonlinear perturbation problem for the semilinear elliptic equation $\Delta u + p(x) u + \phi(x,u) = 0$ in an exterior domain of $\mathbb{R}^N$ with $N \ge 3$. The lower limit of the nonlinear perturbed term $\phi(x,u)$ is given for all nontrivial solutions to be oscillatory. The tools for obtaining our theorems are the so-called ``supersolution-subsolution'' method and some results concerning the oscillation and nonoscillation of solutions of the ordinary differential equation associated with the elliptic equation. A simple example is given to illustrate the main results.