Normalized solutions for <i>p</i> -Laplacian equation with general nonlinearities on bounded domain

We consider the existence, nonexistence and multiplicity of normalized solutions to the $p$ -Laplacian equation on a bounded domain (0.1) \begin{equation}\left\{\begin{array}{ll}-\Delta_p u+\lambda |u|^{p-2}u=g(u),&amp;\text{in }\Omega,\\\int_\Omega |u|^p=\rho, u=0&amp;\text{on }\partial \Omega,\end{array}\right.\end{equation} where $\Omega$ is a bounded domain, $p\geq 2$ . Firstly, under suitable assumptions on $\rho$ , if $g$ is at most mass-critical at infinity, we prove the existence of infinitely many solutions. Secondly, for $\rho$ large, if $g$ is mass-supercritical, we perform a blow-up analysis to show the nonexistence of finite Morse index solutions. At last, for $\rho$ suitably small, combining with the monotonicity argument, we obtain a multiplicity result. In particular, when $p=2$ , we obtain the existence of infinitely many normalized solutions.