We generalize the Thomas-Fermi approach to galaxy structure to include central supermassive black holes and find, self-consistently and non-linearly, the gravitational potential of the galaxy plus the central black hole (BH) system. This approach naturally incorporates the quantum pressure of the fermionic warm dark matter (WDM) particles and shows its full power and clearness in the presence of supermassive black holes. We find the main galaxy and central black hole magnitudes as the halo radius r(h), halo mass M-h, black hole mass M-BH, velocity dispersion sigma, and phase space density, with their realistic astrophysical values, masses and sizes over a wide galaxy range. The supermassive black hole masses arise naturally in this framework. Our extensive numerical calculations and detailed analytic resolution of the Thomas-Fermi equations show that in the presence of the central BH, both DM regimes-classical (Boltzmann dilute) and quantum (compact)-do necessarily co-exist generically in any galaxy, from the smaller and compact galaxies to the largest ones. The ratio R(r) of the particle wavelength to the average interparticle distance shows consistently that the transition, R similar or equal to 1, from the quantum to the classical region occurs precisely at the same point r(A) where the chemical potential vanishes. A novel halo structure with three regions shows up: in the vicinity of the BH, WDM is always quantum in a small compact core of radius rA and nearly constant density; in the region r(A) < r < r(i) until the BH influence radius r(i), WDM is less compact and exhibits a clear classical Boltzmann-like behavior; for r > r(i), the WDM gravity potential dominates, and the known halo galaxy shows up with its astrophysical size. DM is a dilute classical gas in this region. As an illustration, three representative families of galaxy plus central BH solutions are found and analyzed: small, medium and large galaxies with realistic supermassive BH masses of 10(5)M(circle dot), 10(7)M(circle dot) and 10(9)M(circle dot), respectively. In the presence of the central BH, we find a minimum galaxy size and mass M-h(min) similar or equal to 10(7)M(circle dot), larger (2.2233 x 10(3) times) than the one without BH, and reached at a minimal non-zero temperature Tmin. The supermassive BH heats up the DM and prevents it from becoming an exactly degenerate gas at zero temperature. Colder galaxies are smaller, and warmer galaxies are larger. Galaxies with a central black hole have large masses M-h > 10(7)M(circle dot) > M-h(min); compact or ultracompact dwarf galaxies in the range 10(4)M(circle dot) < M-h < 10(7) M-circle dot cannot harbor central BHs. We find novel scaling relations M-BH = DMh3/8 and r(h) = CMBH4/3, and show that the DM galaxy scaling relations M-h = b Sigma(0)r(h)(2) and M-h = a sigma(4)(h)/Sigma(0) hold too in the presence of the central BH, Sigma(0) being the constant surface density scale over a wide galaxy range. The galaxy equation of state is derived: pressure P(r) takes huge values in the BH vicinity region and then sharply decreases entering the classical region, following consistently a self-gravitating perfect gas P(r)=sigma(2)rho(r) behavior.
