Sun–Wong type theorems for second order damped elliptic equations

Abstract

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 = 1 N D i [ a i j ( x ) D j y ] + ∑ i = 1 N b i ( x ) D i y + C ( x , y ) = e ( x ) . When N = 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 ⩾ 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.