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 $${\mathrm{\Delta }}^{2}u-\beta \mathrm{\Delta }u=g(u)\text{in }{\mathbb{R}}^N$$ in the positive and zero mass regimes: in the former, $N\ge 2$ and $\beta > -2\sqrt{m}$,where$m>0$dependson$g$;inthelatter,$N\geq 3$and$\beta>0$. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about $g$; if $N=2$ in the positive mass case, or $N=4$ in the zero mass case, we need to strengthen such assumptions. Our approach is variational.