Determining an unbounded potential for an elliptic equation with a power type nonlinearity

Abstract

In this article we focus on inverse problems for a semilinear elliptic equation. We show that a potential $q$ in $L^{n/2+\epsilon }$, $\epsilon >0$, 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 Hölder 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 $L^{n+\epsilon }$. The authors of arXiv:2202.05290 [math.AP] proved this to be true for Hölder continuous potentials.

11 pages, edited title, added a note on the proofs of theorems 1.1, 1.2 and 1.3 on page 4