In this paper, we establish multiplicity of normalized ground-state solutions for the following mixed local and nonlocal Laplacian equation with general nonlinearity in cylindrical domain \(\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _yu+(-\Delta _z)^s_{a(\epsilon z)}u =\lambda u +h(\epsilon z, u),\ \ (y,z)\in \Omega ,\\ u=0,\ \ (y,z)\in \partial \Omega ,\\ \displaystyle \int \limits _{\Omega }|u|^2dx=\rho ^2, \end{array}\right. } \end{aligned}\) where \(0<s<1\) , \(1\le K<N\) , \(\epsilon >0,\lambda \in {\mathbb {R}}\) , \(\rho >0\) , \(\Omega =\Omega ^\prime \times {\mathbb {R}}^{N-K}\) , \(\Omega ^\prime \subset {\mathbb {R}}^K\) is a smooth bounded domain containing zero, \(\Delta _y\) is the Laplacian with respect to variable \(y\in \Omega \) , \((-\Delta _z)^s_{a(\epsilon \cdot )}\) is the weighted s-fractional Laplacian with respect to variable \(z\in {\mathbb {R}}^{N-K}\) and weight \(a:{\mathbb {R}}^{N-K}\times {\mathbb {R}}^{N-K}\rightarrow {\mathbb {R}}\) , \(h:{\mathbb {R}}^{N-K}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function and may satisfy Sobolev critical or supercritical growth conditions. The existence of normalized solutions is established through variational methods on a truncated problem, ensuring compactness and critical point existence. Multiplicity arises from applying the Lusternik–Schnirelmann category theory to the solution manifold, yielding multiple ground states as the parameter \(\epsilon \) is small enough. The novelties of this paper are listed as follows. (1) We improve the existing results concerning normalized solutions in the literature by establishing refined multiplicity criteria through variational analysis, and generalizing the assumptions on the nonlinear term. Our results are new even if \(K=N\) . (2) The operator \(-\Delta _y+(-\Delta _z)^s\) is anisotropic in \(y\in \Omega ^\prime \) and \(z\in {\mathbb {R}}^{N-K}\) directions, which causes some difficulties in discussing the compactness of energy functional.
