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

We study the existence of normalized ground states for the 3D dipolar Bose-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} -\Delta u+\beta u+\lambda_1|u|^2 u-|u|^4u=0\tag{CQNLS}, \end{align} which has been recently 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 state solution with mass $c$ and for $\lambda_1\geq 0$, (\ref{2}) has no non-trivial solution. We show that by adding a dipole-dipole interaction to (\ref{2}), the geometric nature of (\ref{2}) changes dramatically and techniques as the ones from [31] cannot be used anymore to obtain similar results. More precisely, due to the axisymmetric nature of the dipole-dipole interaction potential, the energy corresponding to (\ref{1}) is not stable under symmetric rearrangements, hence conventional arguments based on the radial symmetry of solutions are inapplicable. We will overcome this difficulty by appealing to subtle variational and perturbative methods and prove the following: (i) If the pair $(\lambda_1,\lambda_2)$ is unstable and $\lambda_1<0$, then for 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, any non-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.