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We study the existence of normalized ground states for the 3D dipolarBose-Einstein condensate equation with attractive three-body interactions:\begin{align}\label{1} -\Delta u+\beta u+\lambda_1|u|^2 u+\lambda_2(K*|u|^2)u-|u|^4u=0.\tag{DBEC} \end{align} When $\lambda_2=0$ or $u$ is radial,(\ref{1}) reduces to the cubic-quintic NLS \begin{align}\label{2} -\Deltau+\beta u+\lambda_1|u|^2 u-|u|^4u=0\tag{CQNLS}, \end{align} which has beenrecently studied by Soave in [31]. In particular, it was shown that for any$\lambda_1<0$ and $c>0$, (\ref{2}) possesses a radially symmetric ground statesolution with mass $c$ and for $\lambda_1\geq 0$, (\ref{2}) has no non-trivialsolution. We show that by adding a dipole-dipole interaction to (\ref{2}), thegeometric nature of (\ref{2}) changes dramatically and techniques as the onesfrom [31] cannot be used anymore to obtain similar results. More precisely, dueto the axisymmetric nature of the dipole-dipole interaction potential, theenergy corresponding to (\ref{1}) is not stable under symmetric rearrangements,hence conventional arguments based on the radial symmetry of solutions areinapplicable. We will overcome this difficulty by appealing to subtlevariational and perturbative methods and prove the following:  (i) If the pair $(\lambda_1,\lambda_2)$ is unstable and $\lambda_1<0$, thenfor any $c>0$, (\ref{1}) has a ground state solution with mass $c$.  (ii) If the pair $(\lambda_1,\lambda_2)$ is unstable and $\lambda_1\geq 0$,then there exists some $c^*=c^*(\lambda_1,\lambda_2)\geq 0$ such that for all$c>c^*$, (\ref{1}) has a ground state solution with mass $c$. Moreover, anynon-trivial solution of (\ref{1}) in this case must be non-radial.  (iii) If the pair $(\lambda_1,\lambda_2)$ is \textit{stable}, then (\ref{1})has no non-trivial solutions.

Normalized ground states for 3D dipolar Bose-Einstein condensate with attractive three-body interactions