  In this article we focus on inverse problems for a semilinear elliptic
equation. We show that a potential $q$ in $L^{n/2+\varepsilon}$,
$\varepsilon>0$, can be determined from the full and partial
Dirichlet-to-Neumann map. This extends the results from [M. Lassas, T.
Liimatainen, Y.-H. Lin, and M. Salo, Partial data inverse problems and
simultaneous recovery of boundary and coefficients for semilinear elliptic
equations, Rev. Mat. Iberoam. (2021)] where this is shown for H\"older
continuous potentials. Also we show that when the Dirichlet-to-Neumann map is
restricted to one point on the boundary, it is possible to determine a
potential $q$ in $L^{n+\varepsilon}$. The authors of arXiv:2202.05290 [math.AP]
proved this to be true for H\"older continuous potentials.
