Oscillation criteria for second order forced elliptic differential equations with mixed nonlinearities

Abstract

In the case of oscillatory potentials, we establish some oscillation criteria for the second order forced elliptic differential equation with mixed nonlinearities div ( A ( x ) ∇ y ) + 〈 b ( x ) , ∇ y 〉 + C ( x , y ) = e ( x ) . When N = 1 , our results extend those of Kong [Q. Kong, Interval criteria for oscillation of second order linear ordinary differential equations, J. Math. Anal. Appl. 229 (1999) 258–270] for certain second order linear ordinary differential equations, Sun and Meng [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second order differential equations with mixed nonlinearities, Appl. Math. Comput. (2007), doi:10.10.1016/j.amc.2007.08.042 ] for certain second order differential equations with mixed nonlinearities, Yang [Q. Yang, Interval oscillation criteria for a second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 135 (2003) 49–64] for a forced second order superlinear ordinary differential equation. When N ≥ 2 , our theorems are more general, and improve the oscillation criteria of Zhuang [R.-K. Zhuang, Annual oscillation criteria for second order nonlinear elliptic differential equations, J. Comput. Anal. Math. (2007), doi:10.1016/j.cam.2007.06.031 ] for a forced second order undamped elliptic differential equation.