Normalized Multi-peak Solutions to Nonlinear Elliptic Problems

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