An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds

Given a conformally transversally anisotropic manifold (M, g), we consider the semilinear elliptic equation

$$\begin{aligned} (-\Delta _{g}+V)u+qu^2=0\quad \hbox { on}\ M. \end{aligned}$$

We show that an a priori unknown smooth function q can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity \(qu^2\), and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.