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We study inverse source problems associated to semilinear elliptic equationsof the form  \[  \Delta u(x)+a(x,u)=F(x),  \]  on a bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq 2$. We show that itis possible to use nonlinearity to break the gauge symmetry of the inversesource problem for a class of nonlinearities $a(x,u)$. This is in contrast toinverse source problems for linear equations, which always have a gaugesymmetry. The class of nonlinearities include certain polynomials andexponential nonlinearities. For these nonlinearities, we determine both$a(x,u)$ and $F(x)$ uniquely from the associated DN map.  Moreover, for general nonlinearities $a(x,u)$, we show that we can recoverthe derivatives $\partial_u^ka(x,u)$ and the source $F(x)$ up to a gauge.Especially, we recover general polynomial nonlinearities up to a gauge andgeneralize results of [FO20,LLLS20] by removing the assumption that $u\equiv 0$is a solution.

Uniqueness results and gauge breaking for inverse source problems of semilinear elliptic equations