An inverse source problem for a quasilinear elliptic equation

We initiate the study of inverse source problems for quasilinear elliptic equations of the form $$\left\{ \begin{array}{ll} \nabla \cdot (\gamma(x,u,\nabla u) \nabla u) = F & \text{in } \Omega, \\ u = f & \text{on } \partial\Omega, \end{array} \right.$$ where $\Omega \subset \mathbb{R}^n$, $n \geq 2$, is a simply connected bounded domain. We consider the specific nonlinearity $\gamma(x,u,\nabla u) = \sigma(x) + q(x) u$, with $q$ assumed to be known. By exploiting the nonlinearity to break the gauge invariance of the problem, we establish unique recovery of both $\sigma$ and $F$ from the associated Dirichlet-to-Neumann (DN) map under the structural conditions $q$ and $\nabla(\sigma/q)$ are nowhere vanishing in $\overline\Omega$. In the absence of these conditions, in particular in the linear case, we demonstrate that the inverse problem admits a gauge obstructing the uniqueness. We use higher order linearizations to obtain a complicated coupled system for the unknowns. The complexity of this system arises in part from the gauge freedom of the linearized equation, which is new in this context. We solve the system by constructing suitable complex geometric optics solutions and applying the unique continuation principle for nonlinear elliptic systems. We anticipate that the solution method developed here will prove useful in other inverse problems as well.