Normalized solutions to a class of $(2,q)$-Laplacian equationsin the strongly sublinear regime

In this paper, we consider the existence and multiplicity of normalized solutions for the following $(2,q)$-Laplacian equation \begin{equation}\label{Equation1} \left{\begin{aligned} &-Δu-Δ_q u+λu=g(u),\quad x \in $\mathbb{R}$^N, &\int_{$\mathbb{R}$^N}u^2 d x=c^2, \end{aligned}\right. \tag{${\mathcal{E}}_{\lambda }$} \end{equation} where $1<q<N$, ${\mathrm{\Delta }}_q=\mathrm{div}\left(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q-2}\mathrm{\nabla }u\right)$ is the $q$-Laplacian operator, $\lambda$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $g:\mathbb{R}\to \mathbb{R}$ is continuous and the behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\underset{s\to 0}{\lim }g(s)\mathrm{/}s=-\mathrm{\infty }$, which includes the logarithmic nonlinearity $$g(s)=s\log s^2\mathrm{.}$$ 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.