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This article studies the inverse problem of recovering a nonlinearity in anelliptic equation $\Delta u + a(x,u) = 0$ from boundary measurements ofsolutions. Previous results based on first order linearization achieve thisunder a sign condition on $\partial_u a(x,u)$, and results based on higherorder linearization recover the Taylor series of $a(x,u)$ with respect to $u$.We improve these results and show that a general nonlinearity, and not just itsTaylor series, is uniquely determined up to gauge near a fixed solution. Ourmethod is based on constructing a good solution map that locally parametrizessolutions of the nonlinear equation by solutions of the linearized equation.

Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$