We study the following nonlinear Schrödinger equation in bounded domains \(\begin{aligned} \left\{ \begin{array}{ll} - \left( {a + b\int _\Omega {|\nabla u{|^2}dx} } \right) \Delta u = \lambda u + \mu (x){|u|^{p - 2}u} & \quad {\textrm{in}}\; \Omega ,\\ \int _{\Omega } {|u|^2dx} = c,\; u = 0 & \quad {\textrm{on}}\; \partial \Omega . \end{array} \right. \end{aligned}\) In the case of \(b=0\) , \(a=1\) and \(\mu (x)=1\) , existence of the solutions with prescribed mass was obtained (see Noris et al. in Anal PDE 7:1807–1838, 2014, Pierotti and Verzini in Calc Var 56:133, 2017). For the general case, in particular sign-changing \(\mu (x)\) , similar existence results to the above problem is still unknown. Here we focus on these unknown case and get sharp existence results via new ideas. When \(2 + \frac{8}{N}< p < 2^*\) , we find constants \(c^*> c_* >0\) , and prove the existence of at least two normalized solutions for \(0<c<c_*\) and at least one normalized solution for \(c_*< c <c^*\) . In particular, when \(\mu (x)\) is a positive constant, the existence of any \(k \in \mathbb {N}^+\) normalized solutions is proved. When \(2 < p \le 2 + \frac{8}{N}\) , we search for a global minimizer of energy functional as normalized solution to the equation, where three cases of \(\mu (x) \ge 0\) , \(\mu (x) \le 0\) and \(\mu (x)\) changing sign are analyzed. Finally, if \( \mu \) is regarded as an unknown real number, one normalized solution is obtained by the analysis of variational problem with double constraints. Tables 1 and 2 will clearly illustrate our main results and the relationship between our work and some related works in the literature.
