Multiple solutions for perturbed quasilinear elliptic problems with oscillatory terms

In this note the existence of multiple non-negative solutions is proved for perturbed quasilinear elliptic problems with oscillatory terms. The equations we studied are equation(PεPε){−Δpu+|u|p−2u=Q(x)(f(u)+εg(u)),x∈RN,u(x)→0,as |x|→∞, where p>1p>1, Q:RN→RQ:RN→R is a continuous positive potential and Q∈L1(RN)∩L∞(RN)Q∈L1(RN)∩L∞(RN). The function f:[0,+∞)→Rf:[0,+∞)→R is a continuous nonlinearity which oscillates at the origin or at infinity and satisfies f(0)=0f(0)=0. The function g:[0,+∞)→Rg:[0,+∞)→R is any arbitrarily continuous function such that g(0)=0g(0)=0. Our conclusions will improve and generalize the results of Kristály (2008) [11], Kristály et al. (2007) [12], and Dai (2009) [13].