On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions

We study the following nonlocal mixed order Gross-Pitaevskii equation $$i\,\partial_t \psi=-\frac{1}{2}\,\Delta \psi+V_{ext}\,\psi+\lambda_1\,|\psi|^2\,\psi+\lambda_2\,(K*|\psi|^2)\,\psi+\lambda_3\,|\psi|^{p-2}\,\psi,$$ where $K$ is the classical dipole-dipole interaction kernel, $\lambda_3>0$ and $p\in(4,6]$; the case $p=6$ being energy critical. For $p=5$ the equation is considered currently as the state-of-the-art model for describing the dynamics of dipolar Bose-Einstein condensates (Lee-Huang-Yang corrected dipolar GPE). We prove existence and nonexistence of standing waves in different parameter regimes; for $p\neq 6$ we prove global well-posedness and small data scattering.