  In the present work we provide a characterization of the ground states of a
higher-dimensional quadratic-quartic model of the nonlinear Schr{\"o}dinger
class with a combination of a focusing biharmonic operator with either an
isotropic or an anisotropic defocusing Laplacian operator (at the linear level)
and power-law nonlinearity. Examining principally the prototypical example of
dimension $d=2$, we find that instability arises beyond a certain threshold
coefficient of the Laplacian between the cubic and quintic cases, while all
solutions are stable for powers below the cubic. Above the quintic, and up to a
critical nonlinearity exponent $p$, there exists a progressively narrowing
range of stable frequencies. Finally, above the critical $p$ all solutions are
unstable. The picture is rather similar in the anisotropic case, with the
difference that even before the cubic case, the numerical computations suggest
an interval of unstable frequencies. Our analysis generalizes the relevant
observations for arbitrary combinations of Laplacian prefactor $b$ and
nonlinearity power $p$.
