Radial and non-radial multiple solutions to a general mixed dispersion NLS equation

We study the following nonlinear Schr\"odinger equation with a forth order dispersion term \[ \Delta^2u-\beta\Delta u=g(u) \quad \text{in } \mathbb{R}^N \] in positive and zero mass regimes: former, $N\geq 2$ $\beta > -2\sqrt{m}$, where $m>0$ depends on $g$; latter, 3$ $\beta>0$. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about if $N=2$ case, or $N=4$ need to strengthen such assumptions. Our approach is variational.