Existence and nonexistence of normalized solutions for nonlinear Schr\"{o}dinger equation involving combined nonlinearities in bounded domain

In this paper, we consider the existence, multiplicity and nonexistence of solutions for the following equation \begin{equation*} \begin{cases} \begin{aligned} &-\Delta u+\omega u=\mu u^{p-1}+u^{q-1},~ u>0 \quad &&\text { in } \Omega, \\ &u=0 &&\text { on } \partial\Omega, \\ \end{aligned} \end{cases} \end{equation*} with prescribed $L^2$-norm $\|u\|_2^2=\rho$, where $N\ge 1$, $\rho>0$, $\mu\in \mathbb{R}$, $1<p\le q$, and $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain. The parameter $\omega\in\mathbb{R}$ arises as a Lagrange multiplier. Firstly, when $2<p\le q\le \frac{2N}{(N-2)^+}$ and $\rho$ is small, we establish the existence of a local minimizer of energy. Furthermore, when $\mu\ge 0$ and $\Omega$ is a star-shaped domain, using the monotonicity trick and the Pohozaev identity, we show that there exists a second solution which is of mountain pass type. Secondly, when $\mu\ge 0$, $N\ge 3$, $1<p\le 2$, $q\ge \max\left\{\frac{2N}{N-2}, 3\right\}$ and $\Omega$ is a convex domain, using the moving-plane method, we prove the nonexistence of normalized solutions for large $\rho$. Finally, when $\mu=0$, $N\ge 3$, $q=\frac{2N}{N-2}$ and $\Omega$ is a ball, we give a dichotomy result of normalized solutions for the Br\'{e}zis-Nirenberg problem by continuation arguments.