Revisiting chameleon gravity - thin-shells and no-shells with   appropriate boundary conditions
  We derive analytic solutions of a chameleon scalar field $\phi$ that couples to a non-relativistic matter in the weak gravitational background of a spherically symmetric body, paying particular attention to a field mass $m_A$ inside of the body. The standard thin-shell field profile is recovered by taking the limit $m_A*r_c \to \infty$, where $r_c$ is a radius of the body. We show the existence of "no-shell" solutions where the field is nearly frozen in the whole interior of the body, which does not necessarily correspond to the "zero-shell" limit of thin-shell solutions. In the no-shell case, under the condition $m_A*r_c \gg 1$, the effective coupling of $\phi$ with matter takes the same asymptotic form as that in the thin-shell case. We study experimental bounds coming from the violation of equivalence principle as well as solar-system tests for a number of models including $f(R)$ gravity and find that the field is in either the thin-shell or the no-shell regime under such constraints, depending on the shape of scalar-field potentials. We also show that, for the consistency with local gravity constraints, the field at the center of the body needs to be extremely close to the value $\phi_A$ at the extremum of an effective potential induced by the matter coupling. 
Takashi Tamaki, Shinji Tsujikawa