  We analyze the thermodynamics of the focusing discrete nonlinear
Schr\"odinger equation in dimensions $d\ge 3$ with general nonlinearity $p>1$
and under a model with two parameters, representing inverse temperature and
strength of the nonlinearity, respectively. We prove the existence of limiting
free energy and analyze the phase diagram for general $d,p$. We also prove the
existence of a continuous phase transition curve that divides the parametric
plane into two regions involving the appearance or non-appearance of solitons.
Appropriate upper and lower bounds for the curve are constructed. We also look
at the typical behavior of a function chosen from the Gibbs measure for certain
parts of the phase diagram.
