Influence of nonlinear perturbed terms on the oscillation of elliptic equations

Yamaoka, Naoto; Sugie, Jitsuro

doi:10.1090/S0002-9939-04-07489-1

Influence of nonlinear perturbed terms on the oscillation of elliptic equations
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by Naoto Yamaoka and Jitsuro Sugie PDF

Proc. Amer. Math. Soc. 132 (2004), 2281-2290 Request permission

Abstract:

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.

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