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We are concerned with the inverse boundary problem of determining anomaliesassociated with a semilinear elliptic equation of the form $-\Delta u+a(\mathbfx, u)=0$, where $a(\mathbf x, u)$ is a general nonlinear term that belongs to aH\"older class. It is assumed that the inhomogeneity of $f(\mathbf x, u)$ iscontained in a bounded domain $D$ in the sense that outside $D$, $a(\mathbf x,u)=\lambda u$ with $\lambda\in\mathbb{C}$. We establish novel uniqueidentifiability results in several general scenarios of practical interest.These include determining the support of the inclusion (i.e. $D$) independentof its content (i.e. $a(\mathbf{x}, u)$ in $D$) by a single boundarymeasurement; and determining both $D$ and $a(\mathbf{x}, u)|_D$ by $M$ boundarymeasurements, where $M\in\mathbb{N}$ signifies the number of unknowncoefficients in $a(\mathbf x, u)$. The mathematical argument is based onmicrolocally characterising the singularities in the solution $u$ induced bythe geometric singularities of $D$, and does not rely on any linearisationtechnique.

Determining anomalies in a semilinear elliptic equation by a minimal number of measurements