the region of $H>0$, they need a smaller norm to compensate the shift of $k$ towards negative values induced by the positive potential, see Eq. (\[stationaryNLSE\]).

Note that the families of states displayed in Figs. \[bifurcation diagrams\](a-c) comprise both $k<0$ and $k>0$, i.e., the leaky and trapped modes alike. In particular, the SSB bifurcation occurs at $k>0$ in panels \[bifurcation diagrams\](a,b), and at $k<0$ in (c) (in the latter case, the SSB sets in at $k=-0.100$, as shown for the same system in Fig. [tails]{}). A noteworthy fact is that the system features the competition of the two different mean-field phase transitions driven by the increase of $N$, i.e., the strength of the self-attraction: the transition from the quasi-bound (leaky) state to the self-trapped one, which was previously found in single-well elevated potentials [@Carr1; @Carr2], and the SSB in the DWP structure. Thus, in the cases shown in Figs. \[bifurcation diagrams\](a,b) the self-trapping transition happens first (at smaller $N$), while in Fig. \[bifurcation diagrams\](c) the SSB takes place prior to the onset of the self-trapping.

The values of the propagation constant and norm at the SSB bifurcation point are shown, as a function of the inner-barrier’s height $A$ \[see Eq. 