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

We study inverse source problems associated to semilinear elliptic equations of the form $$\mathrm{\Delta }u(x)+a(x,u)=F(x),$$ on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$. We show that it is possible to use nonlinearity to break the gauge symmetry of the inverse source problem for a class of nonlinearities $a(x,u)$. This is in contrast to inverse source problems for linear equations, which always have a gauge symmetry. The class of nonlinearities include certain polynomials and exponential 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 recover the derivatives ${\mathrm{\partial }}_u^{k}a(x,u)$ and the source $F(x)$ up to a gauge. Especially, we recover general polynomial nonlinearities up to a gauge and generalize results of [FO20,LLLS20] by removing the assumption that $u\equiv 0$ is a solution.