In this paper, we consider the existence and multiplicity of normalized solutions for the following (2, q)-Laplacian 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 0.2 \(\begin{aligned} g(s)= s \log s^2. \end{aligned}\) 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.
