We study inverse source problems associated to semilinear elliptic equations of the form Δux+ax,u=Fx on a bounded domain Ω⊂Rn , n⩾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 the unique recovery of the source. The class of nonlinearities for which we can uniquely recover the source and nonlinearity, 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 ∂uka(x,u) up to a gauge symmetry. We recover general polynomial nonlinearities up to the gauge symmetry. Our results also generalize results of Feizmohammadi and Oksanen (2020 J. Differ. Equ. 269 4683–719), Lassas et al (2020 Rev. Mat. Iberoam. 37 1553–80) by removing the assumption that u≡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.

中文翻译：

---

半线性椭圆方程反源问题的唯一性结果

我们研究与半线性椭圆方程相关的逆源问题 Δ你X+AX,你=FX 在有界域上 Ω⊂右n , n⩾2 。我们证明可以使用非线性来恢复源F和非线性 A（X,你） 对于一类非线性同时且唯一。这与线性方程的逆源问题形成对比，线性方程总是具有自然（规范）对称性，阻碍了源的唯一恢复。我们可以唯一地恢复源和非线性的非线性类别，包括我们表征的一类多项式和指数非线性。对于一般非线性 A（X,你） ，我们恢复源F（X) 和泰勒系数 ∂你kA（X,你） 达到规范对称性。我们恢复一般多项式非线性直至规范对称性。我们的结果还概括了 Feizmohammadi 和 Oksanen (2020J. 不同。等式。 2694683–719), 拉萨斯等人(2020年马特牧师。伊比利亚人。 371553–80）通过删除以下假设 你==0 是一个解决方案。为了证明我们的结果，我们考虑了围绕可能的大型解决方案的线性化。我们的结果可以带来新的实际应用，因为我们表明许多实际模型不存在限制线性模型逆源问题的适用性的独特恢复的障碍。