  In this paper, we consider the existence and multiplicity of normalized
solutions for the following $(2, q)$-Laplacian equation
\begin{equation}\label{Equation} \left\{\begin{aligned} &-\Delta u-\Delta_q
u+\lambda u=g(u),\quad x \in \mathbb{R}^N, &\|u\|_2^2 =c^2,
\end{aligned}\right. \tag{$\mathscr E_\lambda$} \end{equation} where $1<q<N$,
$\Delta_q=\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right)$ is the
$q$-Laplacian operator, $\lambda$ is a Lagrange multiplier and $c>0$ is a
constant. The nonlinearity $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous
and the behaviour of $g$ at the origin is allowed to be strongly sublinear,
i.e., $\lim \limits_{s \rightarrow 0} g(s) / s=-\infty$, which includes the
logarithmic nonlinearity $$g(s)= s \log s^2.$$ We consider a family of
approximating problems that can be set in $H^1\left(\mathbb{R}^N\right)\cap
D^{1, q}\left(\mathbb{R}^N\right)$ and the corresponding least-energy
solutions. Then, we prove that such a family of solutions converges to a
least-energy solution to the original problem. Additionally, under certain
assumptions about $g$ that allow us to work in a suitable subspace of
$H^1\left(\mathbb{R}^N\right)\cap D^{1, q}\left(\mathbb{R}^N\right)$, we prove
the existence of infinitely many solutions of the above $(2, q)$-Laplacian
equation.
