Normalized solutions for p -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) {equation}\{{array}{ll}-Δ_p u+λ |u|ᵖ⁻²u=g(u),&in Ω,\\∫_Ω |u|^p=ρ, u=0&on ∂ Ω,{array}.{equation} where Ω is a bounded domain, p≥ 2 . Firstly, under suitable assumptions on ρ , if g is at most mass-critical at infinity, we prove the existence of infinitely many solutions. Secondly, for ρ large, if g is mass-supercritical, we perform a blow-up analysis to show the nonexistence of finite Morse index solutions. At last, for ρ 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.