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 $$(-{\mathrm{\Delta }}_g+V)u+q{u}^2=0\text{on }M\mathrm{.}$$ 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 semilinear elliptic equation. This extends the previously known results of the works [FO20, LLLS21a]. Our proof is based on analyzing higher order linearizations of the semilinear equation with non-vanishing boundary traces and also the study of interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation.