Sun–Wong type theorems for second order damped elliptic equations

The oscillation criteria of Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549–560] are extended to the forced second order damped elliptic differential equation with mixed nonlinearities∑i,j=1NDi[aij(x)Djy]+∑i=1Nbi(x)Diy+C(x,y)=e(x). When N=1N=1, our results include and improve those of Nasr [A.H. Nasr, Sufficient conditions for the forced super-linear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc. 126 (1998) 123–125] for a forced second order superlinear differential equation, Wong [J.S.W. Wong, Oscillation criteria for a forced second order linear differential equations, J. Math. Anal. Appl. 231 (1999) 235–240] for a forced second order linear differential equation, Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549–560] for a forced second order differential equation with mixed nonlinearities. When N⩾2N⩾2, our results 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. 217 (2008) 268–276] for a forced second order undamped elliptic differential equation.