# Inverse problems for elliptic equations with power type nonlinearities

## TL;DR

This paper presents a novel method using higher order linearizations to solve Calderón type inverse problems for semilinear elliptic equations with power nonlinearities, enabling potential and manifold recovery without complex geometrical optics solutions.

## Contribution

Introduces a new approach for inverse problems in semilinear elliptic equations that works even when linear solutions are unknown, expanding the scope of solvable inverse problems.

## Key findings

- Determines potential and conformal manifold in 2D from nonlinear Dirichlet-to-Neumann map.
- Solves inverse problems on transversally anisotropic manifolds in higher dimensions.
- Simplifies the Calderón problem for certain semilinear equations in Euclidean space.

## Abstract

We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calder\'on problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.