Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities

In this paper, we study {existence and multiplicity} of normalized solutions for the following $(2,q)$-Laplacian equation \begin{equation*}\label{Eq-Equation1} \left{\begin{array}{l} -Δu-Δ_q u+λu=f(u) \quad x \in $\mathbb{R}$^N , \int_{$\mathbb{R}$^N}u^2 d x=c^2, \end{array}\right. \end{equation*} where $1<q<N$, $N\ge 3$, ${\mathrm{\Delta }}_q=\mathrm{div}\left(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q-2}\mathrm{\nabla }u\right)$ denotes the $q$-Laplacian operator, $\lambda$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $f:\mathbb{R}\to \mathbb{R}$ is continuous, with mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution and the existence of a second solution with higher energy.