Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions

In this paper, by adapting the perturbation method, we study the existence and multiplicity of normalized solutions for the following nonlinear Schrödinger equation { − Δ u = λ u + f ( u ) in Ω , B α , ζ , γ u = 0 on ∂ Ω , ∫ Ω u 2 d x = μ , ( P ) α , ζ , γ μ where Ω ⊂ R N ( N ⩾ 1 ) is a smooth bounded domain, μ > 0 is prescribed, λ ∈ R is a part of the unknown which appears as a Lagrange multiplier, the boundary operator B α , ζ , γ is defined by