Uniqueness results for inverse source problems for semilinear elliptic equations

Abstract

We study inverse source problems associated to semilinear elliptic equations of the form
 \[
 \Delta u(x)+a(x,u)=F(x)
 \]
 on a bounded domain $\Omega\subset \R^n$, $n\geq 2$. We show that it is possible to use nonlinearity to recover both the source $F$ and the nonlinearity $a(x,u)$ simultaneously and uniquely for a class of nonlinearities. 
 This is in contrast to inverse source problems for linear equations, which always have a natural (gauge) symmetry that obstructs \tbr{the} unique recovery of the source. The class of nonlinearities for which we can uniquely recover the source and nonlinearity, \tbr{includes} a class of polynomials, which we characterize, and exponential nonlinearities.
 
 For general nonlinearities $a(x,u)$, we recover the source $F(x)$ and the Taylor coefficients $\p_u^ka(x,u)$ up to a gauge symmetry. We recover general polynomial nonlinearities up to \tbr{the gauge} symmetry. Our results \tbr{also} generalize results of \cite{FO19,LLLS2019partial} by removing the assumption that $u\equiv 0$ is a solution. To prove our results, we consider linearizations around possibly large solutions. 
 
 Our results can lead to new practical applications, because we show that many practical models do not have the obstruction for unique recovery that has restricted the applicability of inverse source problems for linear models.