  In the paper, the results of analysis of elastic scattering and breakup
processes in interactions of the $^{11}$Li nucleus with protons are presented.
The hybrid model of the microscopic optical potential (OP) is applied. This OP
includes the single-folding real part, while its imaginary part is derived
within the high-energy approximation (HEA) theory. For the $^{11}$Li$+p$
elastic scattering, the microscopic large-scale shell model (LSSM) density of
$^{11}$Li is used. The depths of the real and imaginary parts of OP are fitted
to the elastic scattering data at 62, 68.4, and 75 MeV/nucleon, being
simultaneously adjusted to reproduce the true energy dependence of the
corresponding volume integrals. The role of the spin-orbit potential is studied
and predictions for the total reaction cross sections are made. Also, the
cluster model, in which $^{11}$Li consists of $2n$-halo and the $^{9}$Li core
having its own LSSM form of density, is adopted. The respective microscopic
proton-cluster OP's are calculated and folded with the density probability of
the relative motion of both clusters to get the whole $^{11}$Li$+p$ optical
potential. The breakup cross sections of $^{11}$Li at 62 MeV/nucleon and
momentum distributions of the cluster fragments are calculated. An analysis of
the single-particle density of $^{11}$Li within the same cluster model
accounting for the possible geometric forms of the halo-cluster density
distribution is performed.
