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 ΩΩ is an exterior domain in RNRN, f(x,y)=c(x)|y|α−1y+c1(x)|y|β−1y+c2(x)|y|γ−1yf(x,y)=c(x)|y|α−1y+c1(x)|y|β−1y+c2(x)|y|γ−1y and β>α>γ>0.β>α>γ>0. It is assumed that A=(aij)N×NA=(aij)N×N is a real symmetric positive definite matrix function, b=(bi)N×1b=(bi)N×1 is a real vector function, aij∈Cloc1+μ(Ω,R), and bi,c,c1,c2,e∈Clocμ(Ω,R) for all i,ji,j, for some μ∈(0,1)μ∈(0,1).Examples are given to illustrate the results.