We consider the following class of p-Laplacian problem where 1<p<N,a,b are positive parameters,q\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}（（\mathrm{1},+\mathrm{\infty })，\text{?}[\mathrm{0},+\mathrm{\infty }))，\text{?}f\in C([\mathrm{0},+\mathrm{\infty }),[\mathrm{0}，+\mathrm{\infty })). By applying the fixed point theorem in cones, the method of upper and lower solutions and topological degree theory, we obtain the existence and multiplicity of positive solutions for the above p-Laplacian problem. -div(| ?u | p-2?u)=q(| x |)f(u), | x |>1,x∈RN,u(x)=b, | x |=1,u(x)→a, | x |→+∞, （P）