On the multiplicity and concentration of positive solutions for a class of quasilinear problems in RNRN

In this paper, we consider the following quasilinear problem {−εpΔpu+M(x)|u|p−2u=f(u),x∈RN,u∈W1,p(RN),u(x)>0x∈RN, where −Δpu=−div(|∇u|p−2∇u)−Δpu=−div(|∇u|p−2∇u) is the pp-Laplacian of uu for 2≤p0ε>0 is a small parameter. Under more general assumptions for the nonlinearity ff, we prove the existence, multiplicity and concentration of positive solutions for the above problem. Furthermore, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.