Multiple normalized solutions for elliptic problems with critical Hardy–Sobolev nonlinearity in bounded domains

The aim of this paper is to investigate the existence and multiplicity of normalized solutions for the following zero Dirichlet problem of [Formula: see text]-Laplace equation with Hardy–Sobolev nonlinearity [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is a Lagrange multiplier, [Formula: see text] satisfying [Formula: see text]-mass critical or supercritical conditions when [Formula: see text], and [Formula: see text] satisfies exponential critical growth condition when [Formula: see text]. In the [Formula: see text]-mass critical and supercritical cases, the existence of normalized ground state solutions is considered by using the constrained minimization method. In particular, together with the truncation technique, the concentration-compactness principle and the genus theory, we prove that the existence and multiplicity of the normalized solutions in the Sobolev critical case and the exponential critical case, respectively. This paper complements some existence results of normalized solutions for elliptic equations in bounded domains. The innovation point of this paper is that we obtain multiplicity of normalized solutions for [Formula: see text]-Laplace equations with critical Hardy–Sobolev nonlinearity in bounded domains.