Uniqueness results for inverse source problems for semilinear elliptic equations

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.


Introduction
Let Ω ⊂ R n be a bounded domain with C ∞ -smooth boundary ∂Ω with n ⩾ 2. In this paper we consider semilinear elliptic equations of the form ∆u + a (x, u) = F in Ω, u = f on ∂Ω, (1.1) where a = a(x, z) : Ω × R → R is C ∞ -smooth in the z-variable.For presentational purposes we also assume that a (x, 0) = 0 in Ω. (1.2) This condition is not a restriction of generality as it can be achieved by redefining the source F in (1.1).Let us assume for now that the boundary value problem (1.1) is well-posed on an open subset N ⊂ C 2,α (∂Ω).In this case, the Dirichlet-to-Neumann map (DN map) is defined by the usual assignment Here ν denotes the unit outer normal on ∂Ω.In theorem 2.1 we show that if there exists such that the equation (1.1) admits a solution u 0 ∈ C 2,α (Ω) with u 0 | ∂Ω = f 0 , and 0 is not a Dirichlet eigenvalue of ∆ + ∂ z a (x, u 0 ) in Ω, then there is an open neighborhood N ⊂ C 2,α (∂Ω) of f 0 where (1.1) is well-posed in the following sense: For each f ∈ N there exists a solution u f to (1.1) with u f | ∂Ω = f and the solution u f is unique in a fixed neighborhood of u 0 ∈ C 2,α (Ω).If the sign condition ∂ z a (x, z) ⩽ 0 holds for x ∈ Ω and z ∈ R, the assumptions of theorem 2.1 will be satisfied and the DN map is well-defined by [GT83].If F vanishes on Ω, one can take f 0 ≡ 0 on ∂Ω.In this case, theorem 2.1 reduces to similar well-posedness theorems in the literature, such as the one in [LLLS20].
• Inverse source problem: What can we determine about both a and F from the knowledge of the corresponding DN map Λ a,F ?
For general nonlinearities a(x, z) it is impossible to determine both a(x, z) and F(x) simultaneously from the corresponding DN map.This is due to an inherit gauge invariance of the problem, which we will explain later.For inverse source problems of related linear equations, where the aim is to determine a source function from boundary measurements, the gauge invariance of the problem is well-known: Remark 1.1.Let us consider the inverse source problem for the linear equation ∆u + qu = F in Ω, u = f on ∂Ω. (1.4) In this inverse problem one asks if the DN map Λ F : C ∞ (∂Ω) → C ∞ (∂Ω) associated to the above equation determines F uniquely.We assume here for simplicity that the potential function q is known.In general, the answer to the question is negative due to the following observation.Let u solve (1.4) and let ψ be an arbitrary C 2 -function satisfying Let us also define ũ := u + ψ. (1.5) Consequently, we have (ũ| ∂Ω , ∂ ν ũ| ∂Ω ) = (u| ∂Ω , ∂ ν u| ∂Ω ), and ∆ũ + qũ =∆ (u + ψ) + q (u + ψ) =F − qu + ∆ψ + qu + qψ =F + ∆ψ + qψ. (1.6) Hence u and ũ solve the equations ∆u + qu = F and ∆ũ + qũ = F + ∆ψ + qψ respectively.Since u and ũ also have the same Cauchy data on ∂Ω, it follows that the corresponding DN maps are the same: Λ F (f) = Λ F+∆ψ +qψ (f) on ∂Ω.It is thus not possible to determine a source function uniquely from the DN map.
In this work, we consider different types of nonlinearities, including general ones.For general nonlinearities a(x, z), we prove in theorem 1.3 that the corresponding DN map determines the quantities ∂ k z a (x, u 0 (x)) , x ∈ Ω, k ∈ N. (1.7) Here u 0 is a solution to (1.1) corresponding to a boundary value f 0 .As already evidenced by remark 1.1, it might not be possible to recover u 0 from the DN map.This means that in general the condition (1.7) does not determine a(x, z), or even its derivatives in the variable z.Due to the above obstruction to determining a(x, z), and consequently F(x), in general, we mainly focus on nonlinearities a(x, z) of the following special types: • General polynomial nonlinearity: (1.8) • Exponential type nonlinearities: a (x, z) = q (x) e z and a (x, z) = q (x) ze z , (1.9) • Sine-Gordon nonlinearity: (1.10) For these nonlinearities, we show that the corresponding inverse source problems are either uniquely solvable or there is a gauge symmetry, which has an explicit form.The fact that there are nonlinearities for which the related inverse source problem is uniquely solvable is in contrast to inverse source problem for linear equations, which always have the gauge symmetry presented in remark 1.1.That is, nonlinearity can make inverse source problems uniquely solvable.Quadratic nonlinearity has a specific form gauge symmetry, which we now derive.For this, let us assume that u solves (1.1), where a(x, z) is as above.Let ψ ∈ C 2 (Ω).We denote by ã(1) , ã(2) and F another set of C ∞ (Ω)-smooth functions corresponding to a quadratic nonlinearity of the form (1.11) and source term for (1.1).These functions can implicitly depend on ψ.Let Λ and Λ be the DN maps corresponding to the coefficients without and with tilde signs respectively.If we define then we have the chain of equivalences which is equivalent to which is also equivalent to which hold in Ω.By using ∆u = −a (1) u − a (2) u 2 + F and equating the powers of u gives the following system in Ω.
(1.12)If the above system is satisfied, then Consequently, if we additionally require that ψ| ∂Ω = ∂ ν ψ| ∂Ω = 0, then the DN maps Λ and Λ are the same.That is, if we change the coefficients a (1) , a (2) , F to ã(1) , ã(2) , F , the DN map is preserved.Thus, it is at best possible to determine coefficients and a source from the DN map up to the gauge conditions (1.12).
• Earlier works.Before going into our results in detail, we discuss earlier related works.The standard approach in the study of inverse problems for nonlinear elliptic equations was initiated in [Isa93].There the author linearized the nonlinear DN map C ∞ (∂Ω) → C ∞ (∂Ω).The linearization reduced the inverse problem of a nonlinear equation to an inverse problem of a linear equation, which the author was able to solve by using methods for linear equations.Later, second order linearizations, where data depends on two independent parameters, were used to solve inverse problems for example in [AZ21, CNV19, KN02, Sun96, Sun10, SU97].We also mention here the work [Isa01] that considers inverse problems for systems of semilinear equations.
For the case F = 0 in Ω in (1.1), equivalent to u ≡ 0 being a solution, inverse problems for semilinear elliptic equations were recently considered in [FO20,LLLS20].The novelty of these works is that instead viewing nonlinearity as an additional complication in the inverse problem, the works used nonlinearity as a beneficial tool.The method of these two works originates from the seminal work [KLU18], where inverse problems for nonlinear equations were studied in Lorentzian spacetimes.By using the method where nonlinearity is used as a tool, inverse problems for nonlinear equations have been solved in cases where the corresponding inverse problems for linear equations are still open.The method is by now usually called the higher order linearization method.
After the works [KLU18, FO20, LLLS20], the literature about inverse problems for nonlinear equations based on the higher order linearization method, has grown substantially.The works [LLLS20, LLLS21, LLST22, KU20b, KU20a, FLL21, HL23] investigated inverse problems for semilinear elliptic equations with general nonlinearities and in the case of partial data.Inverse problems for quasilinear elliptic equations using higher order linearization have been studied in [KKU22,CFK+21,FKU21].The works [CLLO22,Nur22] studied inverse problems for minimal surface equations on Riemannian surfaces and Euclidean domains.We also mention the works [LL22a, Lin22, LL22b, LL19, LO22, LZ21], where inverse problems for semilinear fractional type equations have been studied.
Inverse source problems for linear equations that regard determination of both unknown sources and coefficients have attracted recent interest.Applications of them include the photo/thermo-acoustic tomography [LU15], magnetic anomaly detection [DLL19,DLL20] and quantum mechanics [LLM19,LLM21].In this paper, we are interested in related nonlinear counterparts of the above works considering linear models.Finally, inverse problems of simultaneously recovering for both nonlinearities and initial data have been considered by [LLLZ22] and [LLL21].
In our first result we show that a a general polynomial nonlinearity and a source are determined by the corresponding DN map up to a gauge condition generalizing (1.12).In the following theorem we denote by the usual binomial coefficients.We also include a converse statement to the result.
We remark that in the above theorem it is sufficient that the domain N of the DN maps is any non-empty open subset of C 2,α (∂Ω).Especially N can be arbitrary small in size.The same holds for other results of this paper.We also remark that we could have let N to be finite, but otherwise unknown, in the assumptions of the theorem.That is, N could be initially assumed to be different for the coefficients (a 1 , F 1 ) and (a 2 , F 2 ).The determination result, given by (1.16) and (1.17), is the same also in this case.
We highlight the special cases of quadratic and cubic nonlinearities of the theorem in the following remark.
We note that the condition (1.16) in theorem 1.1 tells that the highest order coefficient of a polynomial nonlinearity is always uniquely determined.We also mention here that cubic nonlinearities appear for example in Gross-Pitaevskii model for Bose-Einstein condensates [PS16].
With theorem 1.1 and remark 1.2 at hand, it is natural to ask if we can obtain strict uniqueness results for both the nonlinear coefficients and sources.This is indeed the case.For example, in view of (1.18), if one knows the linear term a priori, i.e. a (1) 2 , and assume a (2) = 0, then ψ ≡ 0. This means that the gauge symmetry of the inverse source problem breaks in the sense that we unique and simultaneous determination of the nonlinearity and the source.Interestingly, the next result shows that inverse source problems for general polynomial nonlinearities are uniquely solvable if the second to highest order coefficient is known.
Theorem 1.2 (Unique recovery general case).Assume as in theorem 1.1 and adopt its notation.Suppose additionally that a in Ω, and either a Then all the coefficients are uniquely determined: Remark 1.3.Let us consider remark 1.2 (a) and assume additionally that Then also The above remark in particularly states the following.The inverse source problem of recovering F from the DN map of ∆u + qu + u 2 = F, is uniquely solvable, where q is assumed to be known.This is in contrast to the inverse source problem of ∆u + qu = F, which always has a gauge symmetry by remark 1.1, even if q is known.Thus, we have provided examples (theorem 1.2 and remark 1.3), where nonlinearity can be used to break the gauge symmetry in an inverse source problem.
On the other hand, in case it is a priori known that F 1 = F 2 , then we also have a uniqueness result: Corollary 1.4.Let us adopt the notation and assumptions in theorem (1.20) The above corollary in particularly says the following.If we consider an inverse problem for the equation ∆u + qu + u 2 = F, F known, then we can recover the lower order term q from the DN map.
We also study an inverse source problem for general semilinear elliptic equations and do not assume that the nonlinearity is necessarily a polynomial.In fact, we will prove the next theorem before theorem 1.1 for convenience.
Theorem 1.3.Let Ω ⊂ R n be a bounded domain with C ∞ -smooth boundary ∂Ω, n ⩾ 2. For j = 1, 2, let a j ( • , z) ∈ C α (Ω) satisfy the condition (1.2) and assume that a j (x, z) is C ∞ -smooth with respect to the z-variable.Given F j ∈ C 0,α (Ω) for some 0 < α < 1, let Λ aj,Fj be the DN map of Then, for any f 0 ∈ N , we have for any k ∈ N.Here u are the solutions of (1.21) with boundary condition u As a corollary to theorem 1.3, we do case studies of inverse source problems when the nonlinearity of the model is either of exponential type or a(x, z) = q(x) sin(z).Exponential type nonlinearities arise for example in mathematical modeling of combustion (see e.g.[Vol14]).The nonlinearity a(x, z) = q(x) sin(z) corresponds to the sine-Gordon equation.The DN map and inverse problems for the sine-Gordon equation have been considered for example in [BK89,FP12].The models are chosen so to give examples of cases where the inverse source problem is uniquely solvable, or has an explicit gauge symmetry.
Let q and F belong to C 0,α (Ω), and consider the semilinear elliptic equations For the corresponding inverse source problems we assume that both of the above boundary value problems have a solution u 0 for some boundary value f 0 such that 0 is not an eigenvalue of ∆ + ∂ z a(x, u 0 ).In this case, it follows from theorem 2.1 that the DN maps N → C 1,α (∂Ω) are defined on an open subset N ⊂ C 2,α (∂M) as before by Here, u f is the unique solution on a neighborhood of u 0 to either (1.23) or (1.24) depending on which of the two models we are considering.We remark that in the case q ⩽ 0 the above assumptions are satisfied and theorem 2.1 holds for (1.23) by [GT83, theorem 15.12].In this case, also (1.24) has a solution u 0 for a given boundary value f 0 ∈ C 2,α (∂Ω) by [GT83, theorem 15.12].However, to apply theorem 2.1, one still needs to assume that 0 is not an eigenvalue of ∆ + ∂ z a(x, u 0 ).We also remark that if F is assumed to be small enough, the DN maps of (1.23) and (1.24) are well-defined by proposition 2.1 on a neighborhood of the zero boundary value.
For the nonlinearity a(x, z) = q(x)e z , the inverse source problem is not uniquely solvable due to a gauge symmetry.However, if the nonlinearity is q(x)ze z , and q(x) = 0 for x ∈ Ω, the corresponding inverse source problem has a unique solution.
Suppose that there is an open N ⊂ C 2,α (∂Ω) such that the corresponding DN maps Λ aj,Fj of the equation Then we have: Case 1. Gauge symmetry: (1.25) for which either side of the equation is defined.
Case 2. Unique determination: As the second application of theorem 1.3, we consider the inverse source problem for the elliptic sine-Gordon equation.Again, let q and F belong to C α (Ω), and assume that the equation has a solution for some boundary value f 0 ∈ C 2,α (∂Ω) such that 0 is not an eigenvalue of ∆ + ∂ z a(x, u 0 ).Then the equation is well-posed on a neighborhood N ⊂ C 2,α (∂Ω) of f 0 by theorem 2.1.The DN map of (1.27) is again defined by where u f ∈ C 2,α (Ω) is the unique solution to (1.27) on a neighborhood of u f0 .If F is assumed to be small enough, the DN map of (1.24) is well defined by proposition 2.1 on a neighborhood of the zero boundary value.
For the sine-Gordon equation, the inverse source problem is solvable.
Corollary 1.6.Let Ω ⊂ R n be a bounded domain with C ∞ -smooth boundary ∂Ω, n ⩾ 2. Let q ∈ C α (Ω), and suppose additionally that a j (x, z) = q j (x) sin z, (1.28) for j = 1, 2. Suppose that there is an open set N ⊂ C 2,α (∂Ω) such that the corresponding DN maps Λ qj,Fj of the equation Then q 1 = q 2 and F 1 = F 2 in Ω. (1.30) The paper is organized as follows.In section 2, we prove a well-posedness result for semilinear elliptic equations with source terms.Moreover, a local well-posedness result is also given in section 2, and the proof is left in appendix.In section 3, we prove theorem 1.3 and remark 1.2.We prove corollaries 1.5 and 1.6 in section 4.

Preliminaries
In this section, we prove a local well-posedness result for the Dirichlet problem (1.1) on a neighborhood of a given solution.Let 0 < α < 1 and δ > 0 and denote (2.1) Note that when the source function F of the equation ∆u(x) + a(x, u) = F(x) does not vanish, zero function is not a solution to the equation (1.1).This is the main reason why our wellposedness result differs from the usual ones, such as the one in [LLLS21,KU20a].
Theorem 2.1 (Well-posedness).Let Ω ⊂ R n be a bounded domain with C ∞ boundary ∂Ω and n ⩾ 2, and a = a(x, z) Assume also that 0 is not a Dirichlet eigenvalue of ∆ + ∂ z a (x, u 0 ) in Ω. (2.3) Then there are δ > 0 and C > 0 such that for any f ∈ N δ there exists a unique solution u ∈ C 2,α (Ω) of Proof.We use the standard method, which uses the implicit function theorem in Banach spaces to prove the theorem.A similar proof can be found from the work [LLLS21] where the source F is assumed to vanish.We refer to that work for additional details of the arguments used.Let and assume that u 0 solves (2.2).Consider the map Similar to [LLLS21, section 2], one can show that the map Notice that Ψ(0, u 0 ) = (0, 0), where This is a homeomorphism B 2 → B 3 by the condition (2.3), which is guaranteed by wellposedness and Schauder estimates for linear second order elliptic equations.
Using the implicit function theorem in Banach spaces [RR06, theorem 10.6 and remark 10.5] yields that there is δ > 0 and an open ball N δ ⊂ C 2,α (∂Ω) and a C ∞ map S : N δ → B 2 such that whenever f C 2,α (∂Ω) ⩽ δ we have Since S is smooth and S(0) = u 0 , the solution u = S(f) satisfies Moreover, by the uniqueness statement of the implicit function theorem, by redefining δ > 0 to be smaller if necessary, u = S(f) is the only solution to Ψ(f, u) = (0, 0) whenever f C 2,α (∂Ω) ⩽ δ and As in [LLLS21], one can check that the solution operator S : We remark that if a(x, z) satisfies for all x ∈ Ω and z ∈ R, then the conditions of theorem 2.1 are satisfied by [GT83, theorem 15.12].Under the assumptions of theorem 2.1 above, the boundary value problem (1.1) is well-posed in the following sense: There is f 0 ∈ C 2,α (∂Ω) and δ > 0 such that for each f ∈ f 0 + N δ there exists a solution u f to (1.1) with u f | ∂Ω = f.The solution u f is unique on a fixed neighborhood of u 0 ∈ C 2,α (Ω), where u 0 solves (1.1) with boundary value f 0 .In this case the corresponding DN map f 0 + N δ → C 1,α (∂Ω) defined by the assignment f → ∂ ν u f | ∂Ω is welldefined and C ∞ smooth in the Fréchet sense.
Next we give a well-posedness result in the case when the Dirichlet data and the source F are both sufficiently small.We record it to provide an example where the DN map is always defined.Let We have following result, whose the proof is placed in the appendix.
Proposition 2.1.Let Ω ⊂ R n be a bounded domain with C ∞ boundary ∂Ω and n ⩾ 2. Assume that a(x, 0) = 0 and the condition (2.3) hold with u 0 = 0.There are C > 0, ε > 0 and δ > 0 such that for any F ∈ A ϵ and f ∈ N δ , then there is a unique solution u ∈ C 2,α (Ω) of In particular, for a fixed F ∈ A ε , the map Partly due to the concreteness of presentation, we end this section by an example of unique solvability result in dimension two, where the monotonicity method works well due to the Sobolev embedding H 1 (Ω) → L 6 (Ω).We note that in the example, we do not need to assume the existence of a solution u 0 , the source F does not need to be small, the solutions are globally unique and F can have quite low regularity.
It is not hard to see that Tu − F is the Frechét derivative of the energy functional Since Ω ⊂ R 2 , the Sobolev space H 1 (Ω) embeds in L 6 (Ω).Using this fact, one can show that the operator T is bounded, strictly monotone and coercive.Then by applying the classical energy method, the functional E is coercive and weakly lower semicontinuous on H 1 0 (Ω) (for example, see [FK14,theorem 26.11]).Therefore, E is bounded from below and attains its infimum at some function u ∈ H 1 0 (Ω).Thus u is a solution of (2.6).The uniqueness of u is a direct result of the strict monotonicity of T. We refer to [BCP22, theorem 3.1] for more details about this argument.

Uniqueness for polynomial nonlinearities up to gauge invariances
To better convey the main idea of the proof of theorem 1.1 regarding general polynomial nonlinearities, let us first consider the simpler cases of quadratic and cubic nonlinearities.

Quadratic nonlinearity
In the introduction we showed that the inverse source problem for has a gauge invariance given by the gauge conditions (1.12).We show next that these gauge conditions are the only obstruction to uniqueness in the inverse source problem for quadratic nonlinearities.This is remark 1.2.

Proof of remark 1.2 (a).
By assumption there is N ⊂ C 2,α (∂Ω) such that We apply the higher order linearization method to the equation We denote ϵ = (ϵ 1 , ϵ 2 ), which especially means that ϵ = 0 is equivalent to ϵ 1 = ϵ 2 = 0. Below the index j = 1, 2 corresponds to the different sets of coefficients, and an index ℓ = 1, 2 to ϵ ℓ parameters.Let us denote by u (0) j the solution to With the well-posedness holding on a neighborhood N of f 0 , see theorem 2.1, we can differentiate (3.2) with respect to ϵ ℓ , for ℓ = 1, 2. We obtain where for j, ℓ = 1, 2. It also follows from theorem 2.1 that we know the DN maps of the equation (3.4) for j = 1 and j = 2 agree.Thus, by the global uniqueness result for linear inverse boundary value problems (see e.g.[LLLS20, proposition 2.1] or [SU87] for n ⩾ 3 and [Buk08,BTW19] for n = 2), we have 1 + 2a (2) It then follows by uniqueness of solutions to the Dirichlet problem (3.3) that We next derive the equation for the second order linearization of (3.2) at u (0) j .For j = 1, 2, a straightforward computation shows that where We show next that a (2) 2 in Ω.For that, let us consider v (ℓ) to be the solution of where Q is given in (3.5) and g ℓ ∈ H 1/2 (∂Ω) will be chosen later for ℓ = 1, 2. We multiply (3.7) by v (1) .Moreover, by using ∂ ν w 1 = ∂ ν w 2 on ∂Ω, integration by parts yields Here we used w 1 − w 2 = 0 on ∂Ω and (3.6) and (3.7).By using that products of pairs of complex geometrical optics solutions (CGOs) to (3.4) are dense in L 1 (Ω) for n ⩾ 2, we can choose v (1) and v (2) so that we obtain a (2) For the construction of the CGOs, see [SU87].
Next we take also v (1) as a CGO solution and multiply the above identity by yet another CGO solution v (2) with v (2) | ∂Ω = g 2 , one can integrate the above identity to obtain ˆΩ a (2) By applying the density of CGOs again shows that 2 in Ω. (3.9) Let us then define ψ ∈ C 2 (Ω) as the difference 1 in Ω.
Remark 3.1.Note that if the coefficients of quadratic terms vanish, a (1) (2) 2 = 0 in Ω, then (3.13) describes the gauge symmetry of inverse source problem for linear equation discussed in remark 1.1.
We also remark that in the above proof we could have alternatively used Runge approximation argument to show that a (2) 2 after (3.8).Indeed, if x 0 ∈ Ω, there is by Runge approximation (see e.g.[LLS20]) a solution v (1) such that v (1) (x 0 ) = 0. Together with (3.8), and using the above argument for all x 0 ∈ Ω, shows a (2) 2 in Ω. Runge approximation in similar situations were earlier used in [LLLS21].
As discussed in the introduction, if the linear term of a semilinear equation ∆u + a (1) u + a (2) u 2 = F is known (i.e. a (1) is known a priori), then the DN map determines the other coefficients of the equation uniquely.This is remark 1.3.

Cubic nonlinearity
We move on to prove our results about cubic nonlinearities.For j = 1, 2, we let and let us consider the equation Remark 1.2 (b), which we prove in this section shows that the inverse source problems of the above equation has uniqueness property for both coefficients and source up to a gauge.Before proving remark 1.2 (b), let us derive the gauge of the inverse problem.Assume that u 1 solves (3.14) with boundary value 3 and F 2 another set of coefficients and a source, which may depend on ψ.If we denote u 2 = u 1 + ψ, then we have the chain of equivalences where R is the function given by (3.23).Multiplying (3.24) against the solution v of (3.21) and integrating by parts produces the identity ˆΩ a (3) By choosing v (ℓ) (ℓ = 1, 2, 3) and v to be suitable CGO solutions, we conclude via the above integral identity which proves the first relation in (1.19).

Polynomial and general nonlinearity
In order to prove theorem 1.1, where the nonlinearity is a general polynomial, it is convenient to prove theorem 1.3 about general nonlinearities first.
Proof of theorem 1.3.Let N ∈ N. By using the higher order linearization method, let us take the Dirichlet data to be of the form where ϵ ℓ are parameters such that |ϵ ℓ | are sufficiently small, and each f ℓ ∈ C 2,α (∂Ω), for ℓ = 1, . . ., N. We first linearize the equation (1.21) around the solution u (0) j , so that we can have for j = 1, 2, and ℓ = 1, . . ., N. The uniqueness result for the inverse problem for the linear Shrödinger equation yields again that Moreover, via the uniqueness of solutions, we have 2 in Ω, for ℓ = 1, 2, . . ., N. To proceed, the second linearized equation can be derived as where 2 ) in Ω, for ℓ, m = 1, 2, . . ., N. Similar as before, consider a solution v of by multiplying (3.29) by the function v, then an integration by parts formula yields that 2 ) in Ω by utilizing preceding arguments.Furthermore, by considering higher order linearized equations and using an induction argument, similar to the ones in the proofs of [LLLS20, proof of theorem 1.1] and [KU20a, proof of theorem 1.3], it is not hard to show that (1.22) holds for any k ∈ N, where u (0) j are the solutions of (3.28), for j = 1, 2. As N ∈ N was arbitrary, this completes the proof.
We now prove theorem 1.1.
Using theorem 1.3 again, we have Since a j (x, z) is a polynomial in a, this identity is equivalent to After diving by (N − L − 1)! the above reads (3.32) We rewrite (3.32) as a (3.33) We define 1 . (3.34) Then ψ ∈ C 2,α (Ω) and ψ| ∂Ω = ∂ ν ψ| ∂Ω = 0.By using the induction assumption, that (3.30) holds for k = 0, . . ., L, we write the identity (3.33) as a Here the induction assumption was used in the last equality.By using binomial expansion, the above equality is a Here we have defined (3.37) To complete the proof we compare the coefficients of the powers of u (0) 1 of S 1 and S 2 .We first observe that in the term S 1 , the powers of u (0) 1 range from 0 to L + 1.In the tern S 2 , the powers of u (0) 1 range from 1 to L + 1.We split the remaining proof into two cases according to powers of u (0)

. Case 1:
Let us consider the coefficients of the terms u (0) 1 J , J = 1, . . ., L + 1, in S 1 and S 2 .We observe that the coefficient of u Similarly, the coefficient of u (3.39) On the other hand, a direct computation shows that so that (3.38) and (3.39) are the same.

Case 2:
The term S 2 does not contain the zeroth power of u (0) 1 .We express S 1 as where By redefining the summation index of S 0 , we have This proves the induction step.It remains to prove (1.17).
Recall that the nonlinearity a j (x, z) = N k=1 a (k) j z k , for j = 1, 2, then we can write a 1 (x, u On the other hand, one can also express where we used (3.34) and binomial expansion in the above computation.Similar to the computations of Case 1 in preceding arguments, by comparing the orders of the homogeneous parts u Therefore, which shows (1.17).This proves the assertion.
With theorem 1.1 at hand, we can prove theorem 1.2 immediately., the preceding equality yields that ψ = 0 in Ω.Finally, by applying the (1.16) again as k = N, one can prove F 1 = F 2 in Ω, which completes the proof.
We next prove that if the sources F 1 and F 2 are known in theorem 1.1 and remark 1.2, then it is possible to determine the coefficients uniquely.We have the following corollary, which we formulate in terms of the general polynomial nonlinearity.

Case studies of theorem 1.3
In the end of this paper, we study special cases theorem 1.3, which stated that for general nonlinearities ∂ k z a 1 x, u In general, given only the conditions (4.1), it is not clear how explicit relation between the coefficients (a 1 (x, z), F 1 (x)) and (a 2 (x, z), F 2 (x)) in terms of ψ = u (0) 1 one can find.This final section of this paper consider examples where the relation is explicit.

Case 2.
In this case a j (x, z) = q j (x)ze z .Let u This can be achieved by redefining ε, δ > 0 to be smaller if necessary.As in [LLLS21], one can check that the solution operator S : N δ × A ε → C 2,α (Ω) is a C ∞ map in the Fréchet sense.The normal derivative is a linear map C 2,α (Ω) → C 1,α (∂Ω).Thus for a fixed F ∈ A ε , Λ F : f → ∂ ν u f,F , where u f,F solves ∆u f,F + a(x, u f,F ) = F with u f,F | ∂Ω = f, is also a well defined C ∞ map N δ → C 1,α (∂Ω).