  We study various partial data inverse boundary value problems for the
semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb R^n$
by using the higher order linearization technique introduced in [LLS 19, FO19].
We show that the Dirichlet-to-Neumann map of the above equation determines the
Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The
determination of the Taylor series can be done in parallel with the detection
of an unknown cavity inside the domain or an unknown part of the boundary of
the domain. The method relies on the solution of the linearized partial data
Calder\'on problem [FKSU09], and implies the solution of partial data problems
for certain semilinear equations $\Delta u+ a(x,u) = 0$ also proved in [KU19].
  The results that we prove are in contrast to the analogous inverse problems
for the linear Schr\"odinger equation. There recovering an unknown cavity (or
part of the boundary) and the potential simultaneously are long-standing open
problems, and the solution to the Calder\'on problem with partial data is known
only in special cases when $n \geq 3$.
