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