Annulus criteria for mixed nonlinear elliptic differential equations

New oscillation criteria are obtained for forced second order elliptic partial differential equations with damping and mixed nonlinearities of the form ∇ ⋅ ( A ( x ) | ∇ y | α − 1 ∇ y ) + 〈 b ( x ) , | ∇ y | α − 1 ∇ y 〉 + f ( x , y ) = e ( x ) , x ∈ Ω where Ω is an exterior domain in R N , f ( x , y ) = c ( x ) | y | α − 1 y + c 1 ( x ) | y | β − 1 y + c 2 ( x ) | y | γ − 1 y and β > α > γ > 0 . It is assumed that A = ( a i j ) N × N is a real symmetric positive definite matrix function, b = ( b i ) N × 1 is a real vector function, a i j ∈ C loc 1 + μ ( Ω , R ) , and b i , c , c 1 , c 2 , e ∈ C loc μ ( Ω , R ) for all i , j , for some μ ∈ ( 0 , 1 ) . es are given to illustrate the results.