We introduce a method for solving Calderón 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 $n\ge 3$. In the Euclidean case, we show that one can solve the Calderón problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.

我们介绍了一种求解具有幂型非线性的半线性方程组的Calderón型逆问题的方法。该方法基于高阶线性化，在未知相应线性方程的解的情况下，它可以解决某些非线性方程的逆问题。假设了解非线性Dirichlet-to-Neumann映射，我们将同时确定维度2中的势和保形流形以及维度上的横向各向异性流形上的势$ñ\ge 3$。在欧几里得的情况下，我们表明人们可以用一种出乎意料的简单方式解决某些半线性方程式的Calderón问题，而无需使用复杂的几何光学解决方案。