In this paper we study the existence of multiple asymmetric positive solutions for the following symmetric problem:\[{−Δu+(λ−h(x))u=(1−f(x))up,amp;x∈RN,u(x)>0,amp;x∈RN,u∈H1(RN),\begin {cases} -\Delta u+(\lambda -h(x))u=(1-f(x))u^p, & x\in \mathbb {R}^N,\\ u(x)>0, &\quad x\in \mathbb {R}^N,\\ u\in H^1(\mathbb {R}^N), \end {cases}\]whereλ>0\lambda >0is a parameter,h(x)h(x)andf(x)f(x)are nonnegative radially symmetric functions inL∞(RN)L^\infty (\mathbb {R}^N),h(x)h(x)andf(x)f(x)have compact support inRN\mathbb {R}^N,f(x)≤1f(x)\leq 1for allx∈RNx\in \mathbb {R}^N,1>p>+∞1>p>+\inftyforN=1,2N=1,2,1>p>N+2N−21>p>\frac {N+2}{N-2}forN≥3N\geq 3. We prove that for anyk=1,2,…k=1,2,\,\ldots \,, ifλ\lambdais large enough the above problem has positive solutionsuλu_\lambdaconcentrating atkkdistinct points away from the origin asλ\lambdagoes to∞\infty.