  In this work, we consider quasi-one-dimensional Bose-Einstein condensates
(BECs), with spatially varying collisional interactions, trapped in double well
potentials. In particular, we study a setup in which such a 'collisionally
inhomogeneous' BEC has the same (attractive-attractive or repulsive-repulsive)
or different (attractive-repulsive) type of interparticle interactions. Our
analysis is based on the continuation of the symmetric ground state and
anti-symmetric first excited state of the noninteracting (linear) limit into
their nonlinear counterparts. The collisional inhomogeneity produces a
saddle-node bifurcation scenario between two additional solution branches; as
the inhomogeneity becomes stronger, the turning point of the saddle-node tends
to infinity and eventually only the two original branches remain present, which
is completely different from the standard double-well phenomenology. Finally,
one of these branches changes its monotonicity as a function of the chemical
potential, a feature especially prominent, when the sign of the nonlinearity
changes between the two wells. Our theoretical predictions, are in excellent
agreement with the numerical results.
