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 $pin(4,6]$; 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 $p eq 6$ we prove global well-posedness and small data scattering.
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