In this article, we establish the existence of positive multi-peak solutions to the following elliptic problem -Δv+(λ+V(x))v=vpinΩ,v>0inΩ,∫Ωv2dx=ρ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta v+(\lambda +V(x))v=v^p \ {} &{}\text { in } \Omega ,\\ v>0 &{}\text { in }\Omega ,\\ \int _{\Omega }v^2dx=\rho , \end{array}\right. } \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is a bounded smooth domain of RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^N$$\end{document} or the whole space RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^N$$\end{document}, the exponent p satisfies 1<p<N+2N-2\documentclass[12pt]{minimal}
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\begin{document}$$1<p<\frac{N+2}{N-2}$$\end{document} for N≥3\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 3$$\end{document} and p>1\documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document} for N=1,2\documentclass[12pt]{minimal}
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\begin{document}$$N=1,2$$\end{document}. For the case of mass subcritical, mass critical, and mass supercritical, we shall deal with the effect of ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} on the existence of the solution concentrating at k different points, which belong to either ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega $$\end{document} or Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}, or RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^N$$\end{document}.
