In this paper, we study existence and multiplicity of normalized solutions for the following (2, q)-Laplacian equation \(\begin{aligned} \left\{ \begin{array}{l} -\Delta u-\Delta _q u+\lambda u=f(u) \quad x \in \mathbb {R}^N , \\ \int _{\mathbb {R}^N}u^2 d x=c^2, \\ \end{array}\right. \end{aligned}\) where \(1<q<N\) , \(N\ge 3\) , \(\Delta _q=\operatorname {div}\left( |\nabla u|^{q-2} \nabla u\right) \) denotes the q-Laplacian operator, \(\lambda \) is a Lagrange multiplier and \(c>0\) is a constant. The nonlinearity \(f:\mathbb {R}\rightarrow \mathbb {R}\) 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.
