Normalized solution for $p$-Laplacian equation in exterior domain

We are devoted to the study of the following nonlinear $p$-Laplacian Schr\"odinger equation with $L^{p}$-norm constraint \begin{align*} \begin{cases} &-\Delta_{p} u=\lambda |u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quad\Omega,\\ &u=0\quad\mbox{on}\quad \partial\Omega,\\ &\int_{\Omega}|u|^{p}dx=a, \end{cases} \end{align*} where $\Delta_{p}u=\text{div} (|\nabla u|^{p-2}\nabla u)$, $\Omega\subset\mathbb{R}^{N}$ is an exterior domain with smooth boundary $\partial\Omega\neq\emptyset$ satisfying that $\R^{N}\setminus\Omega$ is bounded, $N\geq3$, $2\leq p<N$, $p<r<p+\frac{p^{2}}{N}$, $a>0$ and $\lambda\in\R$ is an unknown Lagrange multiplier. First, by using the splitting techniques and the Gagliardo-Nirenberg inequality, the compactness of Palais-Smale sequence of the above problem at higher energy level is established. Then, exploiting barycentric function methods, Brouwer degree and minimax principle, we obtain a solution $(u,\la)$ with $u>0$ in $\R^{N}$ and $\la<0$ when $\R^{N}\setminus\Omega$ is contained in a small ball. Moreover, we give a similar result if we remove the restriction on $\Omega$ and assume $a>0$ small enough. Last, with the symmetric assumption on $\Omega$, we use genus theory to consider infinite many solutions.