  We consider a lattice equation (Salerno model) combining onsite self-focusing
and intersite self-defocusing cubic terms, which may describe a Bose-Einstein
condensate of dipolar atoms trapped in a strong periodic potential. In the
continuum approximation, the model gives rise to solitons in a finite band of
frequencies, with sechlike solitons near one edge, and an exact peakon solution
at the other. A similar family of solitons is found in the discrete system,
including a peakon; beyond the peakon, the family continues in the form of
cuspons. Stability of the lattice solitons is explored through computation of
eigenvalues for small perturbations, and by direct simulations. A small part of
the family is unstable (in that case, the discrete solitons transform into
robust pulsonic excitations); both peakons and cuspons are stable. The
Vakhitov-Kolokolov criterion precisely explains the stability of regular
solitons and peakons, but does not apply to cuspons. In-phase and out-of-phase
bound states of solitons are also constructed. They exchange their stability at
a point where the bound solitons are peakons. Mobile solitons, composed of a
moving core and background, exist up to a critical value of the strength of the
self-defocusing intersite nonlinearity. Colliding solitons always merge into a
single pulse.
