  The typical AdS/QCD models deal with the large-N_c limit of QCD, as a
consequence the meson spectrum consists of the infinite number of states that
is far from the real situation. Basing on introduction of anharmonic
corrections to the holographic potential, the corrections whose existence has
been recently advocated, we construct a class of bottom-up holographic models
describing arbitrary finite number of states in the sector of light mesons.
Within the proposed approach, the spectrum of masses square has the following
properties: It is linear, $m_n^2\sim n$, at not very large n, nonlinear at
larger n, with the nonlinear corrections being subleading in 1/N_c, has a
limiting mass, and the number of states is proportional to N_c. The considered
holographic models reflect thereby the merging of resonances into continuum and
the breaking of gluon string at sufficiently large quark-antiquark separation
that causes the linear Regge trajectories to bend down. We show that these
models provide a correct description for the spectrum of excited rho-mesons.
