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Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

We study a class of two-dimensional non-linear Schr\"odinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a non-trivial adaptation of the standard properties of Schwartz symmetrisation for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.


Introduction and main results
In this work we study the existence of standing waves and the well-posedness of the two-dimensional, point-like perturbed, singular Hartree equation (1.1) i∂ t ψ = −∆ α ψ + (w * |ψ| 2 )ψ , in the complex-valued unknown ψ ≡ ψ(t, x), (t, x) ∈ [0, ∞) × R 2 , where the convolution potential w is a real-valued measurable function, and −∆ α is the self-adjoint point-like singular perturbation of the ordinary negative Laplacian on L 2 (R 2 )functions, located at the origin x = 0 and with inverse scattering length given by α ∈ R, in suitable units. There are multiple motivations for the recently increasing interest towards nonlinear Schrödinger equations (NLS) of Hartree type like (1.1) (i.e., convolutive, nonlocal), or also with local semi-linearity, with the additional point-like perturbation at one or more distinguished points. On the one hand, such equations are realistic effective models for the time evolution of the density of particles |ψ(t, x)| 2 of a large assembly of identical bosons at ultra-low temperature interacting through a twobody potential w and additionally coupled to strong, delta-like impurities located at fixed points in space -one impurity at x = 0 in the present case. Without the singular perturbation, hence with the Laplacian −∆ in place of −∆ α , or also with variants such as the magnetic or the semi-relativistic Laplacian, a variety of highly sophisticated mathematical techniques is today available to rigorously derive such effective NLS, in the limit of infinitely many particles, from the linear, many-body Schrödinger equation [50,15]. For this apparatus to work, and the same would apply to the point-like, singular-perturbed Laplacian, it is crucial to know that the emerging NLS is well-posed in a convenient regularity space, so as to have suitable a priori norms available that are uniformly conserved in time and can be exploited in the control of the limit of infinitely many particles.
From an analogous perspective, the non-perturbed version of (1.1) has a meaning of effective equation for the quantum plasma dynamics, when considered with the magnetic Laplacian and as part of coupled systems of Maxwell-Schrödinger type, a scenario already well under study both for its rigorous derivation as an effective description from the many-body Pauli-Fierz Hamiltonian [41], and concerning wellposedness and standing waves [23,48,36,24,11]: here too, adding a point-like perturbation is of relevance for a more accurate modelling with impurities.
On the other hand, there is an autonomous relevance of (1.1) per se, as (1.1) poses novel technical problems arising in the study of its various features as a dynamical equation (well-posedness, scattering, ground state), precisely due to the singular point-like perturbed nature of −∆ α , which makes an amount of standard analytic tools from the theory of the linear and non-linear Schrödinger equation not directly applicable.
Indeed, whereas in dimension one the point-like nature of the singularity has the explicit structure −∆ + δ(x), thereby allowing for a much more accurate analysis of (1.1), and its counterparts with local semi-linearity, in terms of local and global well-posedness, blow-up, scattering, asymptotic stability, solitons, standing waves, ground state [6,30,3,5,4,9,14,39,38,27,10,43,44,28], the case of dimension two or three presents a much more uncharted territory, ultimately due to the circumstance that −∆ α is not built as a form perturbation.
This has led only recently to our characterisation [35] of a whole family of fractional powers of the three-dimensional −∆ α and hence of the new 'perturbed' Sobolev spaces adapted to it, whence then the first proof of local and global wellposedness of the three-dimensional counterpart of (1.1) [46]. In the same spirit, the well-posedness of the two-and three-dimensional point-perturbed singular NLS with local semi-linearity has been later established in [20], also outlining perspectives and open problems (on the types of blow-up and scattering). And it is only in the course of the final draft of this work that two significant contributions appeared on the issues of existence and stability or instability of the ground state of the twodimensional point-perturbed singular NLS with local semi-linearity, respectively in the mass sub-critical [2] and the energy sub-critical [33] case. Also the linear flow generated by −∆ α , in two dimensions, has attracted recent attention, significantly with the proof of the L p -boundedness of the wave operator associated to the pair (−∆ α , −∆) [26,25,52], whence also Strichartz and dispersive estimates.
One can also see that on those functions g ∈ D(−∆ α ) vanishing on an open U ⊂ R 2 , one has (−∆ α g) U ≡ 0, and therefore away from the origin −∆ α acts precisely as −∆. The non-trivial interaction with the origin is encoded by a boundary condition of the form (1.2), or equivalent ones: actually, −∆ α is only a perturbation of −∆ in the L 2 -sector of spherically symmetric functions, and with s-wave scattering length equal to (−2πα) −1 . The above stringent interpretation of the contact, point-like nature of the interaction is confirmed by the fact that −∆ α can be also reconstructed as the limit, in the resolvent sense, of ordinary Schrödinger operators −∆ + V ε (x), with potentials V ε on a spatial scale ε −1 that shrink and squeeze to a peaked profile around x = 0 as ε ↓ 0 [7,47].
The expressions (1.2) or (1.6) make sense also when one takes formally α = ∞, and reproduce in this case the ordinary self-adjoint , of which the one with α = ∞ is the Friedrichs extension.
Two natural 'regularity' spaces adapted to −∆ α are respectively, its operator domain H 2 α (R 2 ) := D(−∆ α ) and form domain H 1 α (R 2 ) := D[−∆ α ], referred to also as energy space. As −∆ α is lower semi-bounded, and the bottom of its spectrum is e α , H 1 α (R 2 ) is complete with respect to the energy norm (and a Hilbert space with respect to the scalar product induced by (1.8)), H 2 α (R 2 ) is dense in H 1 α (R 2 ), and (1.9) g 2 By standard structural properties of the form domain of lower semi-bounded self-adjoint extensions of symmetric operators (see, e.g., [34]), one has (1.10) the above sum not depending on the choice of ω (whereas the decomposition of an element g ∈ H 1 α (R 2 ) does), and in the sense of equivalence of norms. We shall require a somewhat restrictive condition on the convolutive potential in the Hartree non-linearity, namely that it has a definite sign (positive, for concreteness) and monotonicity, thus re-writing (1.1) as where the parameter θ = ±1 selects, respectively, the defocusing or focusing behaviour of the non-linearity, and , is non-negative, radial, non-increasing, and not identically zero.
This includes the meaningful cases of pure-power behaviour w(x) = |x| −η , η ∈ (0, 2), and more severe locally-L 1 singularities as well. In fact, an amount of intermediate results of the present work are obtained under more general conditions than (1.13) (this is the case for the analysis of local well-posedness, which, as generally known and as will emerge in Section 5, or also in the above-mentioned recent study [2], is insensitive of the sign definiteness or monotonicity of w); yet, (1.13) is crucial for the technique we employ for the detailed study of the ground state, as further remarked in a moment.
Our two main focuses in this work are the characterisation of standing waves for (1.12) in the focusing case, and the global well-posedness both in the defocusing and focusing case, in the latter scenario covering both the mass sub-critical and the mass critical regimes.
With the above terminology we mean the following. For w ∈ L p1 (R 2 ) + L p2 (R 2 ), with p 1 , p 2 ∈ [1, ∞), and p := min{p 1 , p 2 }, we can prove (Lemma 3.2) (1.14) and we denote by C gn the (Gagliardo-Nirenberg type) optimal constant (dependent on α and w) in (1.14). If p > 1, then the r.h.s. is sub-quadratic in ψ H 1 α and the non-linearity is said mass sub-critical. If p = 1, then the non-linearity is mass critical.
In fact, again when w ∈ L p1 (R 2 ) + L p2 (R 2 ), with p 1 , p 2 ∈ [1, ∞), we shall prove that the Hartree non-linearity is energy sub-critical, explicitly, . Actually, in the mass sub-critical case, the Hartree non-linearity is shown to be even locally Lipschitz in the energy space (see estimate (6.5)). It is therefore natural to study the solution theory for (1.12) in the energy space H 1 α (R 2 ). Concerning our first focus, we recall that standing wave solutions to (1.12) when θ = −1 are of the form e −iλt Q α for some λ ∈ R and some non-identically-zero Observe that (1.16) does make sense, when Q α ∈ H 1 α , at least as an identity in In order to prove the existence of standing waves, we introduce the Weinstein type functional For H 2 α -functions the numerator above reads simply (−∆ α + λ)v, v L 2 . When λ > |e α | the functional (1.17) is actually bounded from below by a positive constant, as a consequence of estimate (1.14). Associated to the Weinstein functional we consider the optimisation problem We will show in Lemma 4.1 that W (w) α,λ is of class C 1 on H 1 α (R 2 )\{0}, and that any ground state v α (i.e., a minimiser to (1.18)) satisfies the Euler-Lagrange equation Observe that W (w) α,λ and Λ (w) α,λ are homogeneous, respectively, of order zero and −2 (i.e., W α,λ (v α ) v α , and for the latter function one has Λ (w) α,λ (Q α ) = 1. This explains that from the solutions to the Euler-Lagrange equation (1.19) the standing waves for (1.12) have precisely the form e −iλt Q α with Q α solving (1.16).
We have the following result on the existence and symmetry of ground states.
Theorem (1.1) implies that every minimiser v α for (1.18) has a non-trivial singular (i.e., proportional to G λ ) component, as immediately follows from the canonical representation (1.22) and the fact that |f α | > 0. In particular, the ground states for α ∈ R are different from the H 2 -ground states of the classical unperturbed Hartree equation.
In particular, Q α belongs to H 2 α (R 2 ), it has a non-trivial singular component, and in view of (1. The main difficulty in order to establish the existence of ground states is to choose a minimising sequence for (1.18) satisfying suitable symmetry properties which survive in the limit. In the classical case α = ∞, thus with the ordinary, unperturbed Laplacian in (1.16), it is the Schwartz symmetrisation that provides a final minimising sequence in H 1 (R 2 ) consisting of spherically symmetric, positive, non-increasing functions. In fact (see, e.g., [40]), the Schwartz symmetrisation f → f * preserves any L p -norm, obeys the Pólya-Szegő inequality as well as the Riesz inequality and precisely (1.25) is crucial in the construction of a minimising sequence for with positive, radial and decreasing H 1 -functions. At finite α, instead, thus with the point-like singular perturbed Laplacian in (1.16)-(1.18), the elements of the adapted Sobolev space H 1 α (R 2 ) have the structure f + c G 1 for f ∈ H 1 (R 2 ) and c ∈ C (see (1.10) above), and a generalisation of the Riesz inequality is needed of the form in order to perform the minimisation argument. We shall establish (1.26) (Lemma 3.1) based on a Brascamp-Friedberg-Lieb-Luttinger inequality.
In addition, an accurate analysis of the equality case in the Pólya-Szegő inequality (1.24), together with the H 2 α -regularity for v α guaranteed by the Euler-Lagrange equation (1.19), allows to show that every ground state is (up to a constant phase) spherically symmetric, strictly positive, and strictly decreasing (Proposition 4.3), thereby concluding the proof of Theorem 1.1.
Our second main focus concerns the local and global well-posedness of (1.12) in the energy space. This field is under a comprehensive and well-established control in the classical, unperturbed case α = ∞, including also when (1.12) contains much more singular and non-symmetric convolution potentials, together with electric and magnetic potentials, possibly depending on time (see, e.g., [21,45,12] and the references therein).
The material announced so far is discussed in Sections 2-4, concerning the characterisation of ground states and the proof of Theorem 1.1, and in Sections 5-7, concerning the local and global well-posedness of (1.29) and the proof of Theorem 1.3.
Notation-wise, we adopt throughout the convention that the L 2 -product ·, · L 2 is linear in the first entry and anti-linear in the second, we write x for √ 1 + x 2 , x ∈ R, we use the short-hand L p T X and W s,p T X for the spaces L p ([0, T ], X ) and W s,p (I, X ) for some Banach space X and T > 0, and we write H s (R d ) for the Sobolev space W s,2 (R d ). By A B, with A, B > 0, we mean as customary that A CB for some constant C, and A κ B indicates that C depends on the parameter κ. A ≈ B stands for A B and B A, and A ≈ κ B has an obvious analogous meaning. The dual exponent p/(p−1) of the index p ∈ [1, +∞] is denoted as usual by p ′ . In identities f = g between measurable functions the 'for almost every x' declaration is tacitly understood. The convention for the Fourier transform shall be ϕ(ξ) = (2π) − d 2 R d e −iξx ϕ(x) dx .

Interpolation and dispersive properties of −∆ α
It is beneficial to exploit certain interpolation and dispersive properties of the two-dimensional operator −∆ α in order to single out relevant facts for the forthcoming analysis: the embedding of the energy space into L p -spaces (Lemma 2.2), the characterisation of the 'singular perturbed' Sobolev space H s α (R 2 ) for intermediate s ∈ (0, 1) (Proposition 2.3), a Gagliardo-Nirenberg type inequality adapted to H 1 α (R 2 ) (Corollary 2.5), and dispersive and Strichartz estimates adapted to −∆ α (Proposition 2.6 and Corollary 2.7).
Next we consider the Sobolev space H s α (R 2 ) for intermediate s ∈ [0, 1). In fact, a standard application of the spectral theorem allows to define the fractional power (ω − ∆ α ) s 2 for arbitrary s ∈ R, whenever ω > |e α |, as well as the associated singular perturbed Sobolev space with norm ) . The case s = 0 corresponds to L 2 (R 2 ), and for positive s the space H −s α (R 2 ) is the topological dual of H s α (R 2 ). Clearly, the explicit representation (2.11) for a generic element of H s α (R 2 ), as well as the norm (2.12), are ω-dependent, all such norms being equivalent for the considered ω > |e α |. In (1.8) the standard H 1 α -norm was fixed by the choice ω = −e α + 1. Again based on the spectral theorem, one recognises H s α (R 2 ), when s ∈ (0, 1), as an interpolation space between L 2 (R 2 ) and H 1 α (R 2 ). With the customary notation [H 1 , H 2 ] θ , θ ∈ (0, 1), for the interpolation space between two Hilbert spaces H 1 and H 2 both embedded continuously in a larger Hilbert space [16], one thus writes This also provides the interpolation inequality What is less evident, instead, is the actual coincidence of this singular perturbed Sobolev space and its classical counterpart.
It is convenient to recall (see, e.g., [16]) that the interpolation space [H 1 , H 2 ] θ consists of those elements h ∈ H 1 ∔ H 2 (as a direct sum of Hilbert spaces) such that h = F (θ) for some F ∈ F (H 1 , H 2 ), where the latter is the space of all bounded and continuous (H 1 ∔ H 2 )-valued functions F on the complex strip which are analytic on the open stripS and furthermore satisfy the property that the two restrictions of F to the boundary of S, namely the functions t → F (it) and t → F (1 + it), are continuous from the real line to, respectively, H 1 or H 2 , and vanish at infinity. Observe that F (H 1 , H 2 ) is a Hilbert space with norm In the present case (see (2.13) ). In particular, owing to the canonical decomposition (1.10) for H 1 α (R 2 ) and to the H 1 α -norm expression (1.11), In order to show now that actually v ∈ H s (R 2 ), let us consider two auxiliary boundary value problems for the Laplace equation in the strip S (recall that a function ϕ ∈ C 2 (U, C) is analytic in the open domain U ⊂ C if and only if it satisfies the Laplace equation (∂ 2 x ϕ)(z) + (∂ 2 y ϕ)(z) = 0 for z ∈ U , where x = Re z and y = Im z). First, we consider the unique solution ϕ to (recall that both F (it) and f ω (t) vanish as |t| → ∞), then we consider the unique solution η to and therefore, setting (observe that we used the inclusion (2.8)). The solution to the latter problem is unique and is precisely F . From Φ ≡ F we then deduce, for s ∈ (0, 1), having used in the last step both ϕ(s) ∈ H s (R 2 ) and (2.6). The claim is thus proved.
A direct useful consequence of Proposition 2.3 and the classical compact Sobolev embedding is the following.
3 also allows to obtain a useful inequality of Gagliardo-Nirenberg type.
Last, we turn to the dispersive properties of the linear propagator (e it∆α ) t∈R . In [26,52] the L p -boundedness was established, for p ∈ (1, ∞), of the wave operators for the pair (−∆ α , −∆) in two dimensions (similar properties have been also investigated in dimension one and three [29,32,37,31]). As a direct consequence, this yields the following dispersive and Strichartz estimates.
(ii) and the Strichartz estimates
Observe that L 1 -L ∞ dispersive estimates cannot hold: for, even a smooth initial datum ψ evolves at later times into a function e it∆α ψ that exhibits, for almost every non-zero t, a non-trivial singular component proportional to G ω ∈ L ∞ (R 2 ).
Given the explicit structure of the absolutely continuous subspace for −∆ α , the above inequalities can be generalised, locally in time, without orthogonal projection.

Existence of ground states
In this Section we prove the existence of a minimiser of the optimisation problem (1.18) claimed in Theorem 1.1.
Some preparation is in order. To begin with, as anticipated in the Introduction, our approach requires a control of the Hartree non-linearity by performing a symmetric re-arrangement of its terms that be compatible with the internal structure ∔ span{G ω } of the energy space. As usual, we denote with ϕ * the Schwartz symmetrisation of a given measurable function ϕ : R 2 → R + [40]. Observe that ϕ → ϕ * is not linear.
Proof of Lemma 3.1. We have to check the validity of (i) In turn, (i) follows once one checks (ii) in combination with the ordinary Riesz inequality (1.25). Now, (ii) follows from the BFLL inequality (3.2) with the special choice N = 5, n = 2, and With an analogous choice one establishes also (iii).
Concerning the Hartree non-linearity, we further need for multiple purposes a standard control of its integrability and local Lipschitz property in suitable L p -sense and H 1 α -sense.
In the preceding two lemmas, the even symmetry of w was not needed. Instead, when such a condition is assumed (un particular, under (1.13)), standard change of variable and Fubini's theorem guarantee the following property that for convenience we single out in a separate lemma. Lemma 3.3. Let w, f, g be measurable functions on R 2 , and assume that w is real-valued and even. Suppose moreover that As a further preparation, let us provide a more explicit connection between the equivalent expressions (1.8)-(1.9) and (1.11) for the H 1 α -norm. Regarding, as mentioned, −∆ α as a self-adjoint extension of a symmetric operator whose Friedrichs extension is the self-adjoint −∆, and taking into account that the Friedrichs and the α-extension differ, in the resolvent sense, by the rank-one projection onto G ω (see (1.6)), one has the following Birman formula (see, e.g., [34,Theorem 7]) for the quadratic form of −∆ α evaluated on a generic element , c ∈ C, and ω > 0: .
Next, for the minimisation (1.18) we show that one can select a minimising sequence with certain explicit features as follows.
Lemma 3.4. Let α ∈ R and λ > |e α |. Assume that w satisfies the condition (1.13). The minimisation problem (1.18) admits a minimising sequence (v n ) n∈N in H 1 α (R 2 ) such that, for every n, (3.13) v n = f n + c n G λ for some c n 0 and some non-negative, radial function f n ∈ H 1 (R 2 ) that is monotone decreasing in |x|, and additionally satisfies (3.14) Such a sequence is uniformly bounded in H 1 α (R 2 ). Proof. Any minimising sequence (v n ) n∈N has the structure v n = f n + c n G λ with f n ∈ H 1 (R 2 ) and c n ∈ C, on account of (1.10). For any such (v n ) n∈N , the new sequence ( v n ) n∈N defined by v n := f n + c n G λ , with f n := |f n | and c n := |c n | having used (3.11) and |∇|f n || |∇f n |. Thus, showing that ( v n ) n∈N too is a minimising sequence. The further sequence (v n ) n∈N defined by v n := f n + c n G λ , f n := ( f n ) * (namely, with the sole Schwartz symmetrisation of the regular part of each v n ) still belongs to H 1 α (R 2 ), since ( f n ) * ∈ H 1 (R 2 ), owing to the invariance of the L 2norm under symmetrisation and the Pólya-Szegő inequality (1.24). Moreover, by construction, the coefficient c n of the singular part is non-negative, and the regular part f n is non-negative, radial, and monotone decreasing in |x|. Lemma 3.1, the Pólya-Szegő inequality again, and the spherical symmetry of w, imply α,λ ( v n ) . Thus, (v n ) n∈N too is a minimising sequence. An obvious multiplicative re-sizing of v n allows to match the additional condition (3.14). We have so far shown that a minimising sequence can be indeed chosen with all the properties stated in the Lemma, but for the uniform H 1 α -boundedness, which we prove now. Since v n α,λ , having used (3.12) with ω = λ 1 − e α and (3.14), (v n ) n∈N is indeed bounded in H 1 α (R 2 ).
For the interest per se and for later application (Proposition 4.3), let us single out the key property emerged from the proof of Lemma 3.4: Corollary 3.5. Let α ∈ R and λ > |e α |. Assume that w satisfies the condition (1.13). Given a non-zero element f +c G λ ∈ H 1 α (R 2 ), according to the decomposition (1.10), then also |f | + |c| G λ and |f | * + |c| G λ belong to H 1 α (R 2 ) \ {0} (where |f | * is the Schwartz symmetrisation of |f |), and For the remaining part of this Section, (v n ) n∈N shall denote a minimising sequence with the features given by Lemma 3.4. In particular, Because of the uniform H 1 α -boundedness of (v n ) n∈N , there exists (3.17) v , possibly a null function, such that v n ⇀ v • H 1 α -weakly, which in turn implies that f n ⇀ f • weakly in H 1 (R 2 ) and c n → c • . Moreover, f • is necessarily nonnegative, spherically symmetric, and radially decreasing, and c • 0. This, and the properties of G λ , imply that v • too is non-negative, spherically symmetric, and radially decreasing.
The next ingredient is a compactness result for the Hartree type non-linearity.
Lemma 3.6. Let α ∈ R. Assume that w satisfies the condition (1.13) and that (g n ) n∈N is a uniformly bounded sequence in H 1 α (R 2 ) consisting of non-negative, spherically symmetric functions and such that g n ⇀ g • H 1 α -weakly. Then, up to subsequence, Proof. Clearly, g • is non-negative and spherically symmetric too. From and from (3.7) one finds As 4p 2p−1 ∈ (2, 4] ⊂ (2, ∞), the compact embedding H 1 α,rad (R 2 ) ֒→ L 4p 2p−1 (R 2 ) from Lemma 2.2 is applicable; using this fact, and the uniform H 1 α -boundedness, one concludes that up to extracting a sub-sequence, the expression above converges to 0 as n → ∞. Proof. The constraint (3.14) from Lemma 3.4, and Lemma 3.6, imply that, up to re-fining the minimising sequence (v n ) n∈N , which prevents v • from being almost-everywhere vanishing.
We can finally assemble the above preparations and address the core part of the proof of the existence of ground states (Theorem 1.1).
We saw that, as n → ∞, with f • being non-negative, spherically symmetric, and radially decreasing. Moreover v • satisfies the constraint (3.20) Since f n ⇀ f H 1 -weakly, then both ∇f n ⇀ ∇f • and f n ⇀ f • L 2 -weakly, and the weak lower semi-continuity of L 2 -norms implies On the other hand, since W i.e., f n → f • in H 1 (R 2 ). Together with c n → c • , and in view of (1.11), this means also v n → v • in H 1 α (R 2 ). As a consequence, that is, v • is a minimiser. And since v • is non-zero, then W (w) α,λ > 0. The existence part of Theorem 1.1 is thus established. For this ground state v • also the nonnegativity, the spherical symmetry, and the radial decreasing are established, as well as the analogous properties for the regular part f • of v • .

Structural properties of the ground states
We come now to characterising the structural properties of ground states claimed in Theorem 1.1, namely the real positivity (up to constant multiplicative phases), the H 2 α -regularity, the spherical symmetry, and the strict radial decrease. We first obtain the explicit Euler-Lagrange equation associated to W Proof. Let us set L := −∆ α +λ, which is a positive, self-adjoint operator on L 2 (R 2 ), with form domain H 1 α (R 2 ). A direct computation using (1.9) yields Given v, h ∈ H 1 α (R 2 ), a direct computation (using Lemma 3.3) and the bound (3.8) yield α,λ is of class C 1 on H 1 α (R 2 ) \ {0}, and the Leibniz rule, together with (a), (b), and the identity Λ Suppose now that v α is a minimiser for the problem (1.18). Then v α is a critical point for the functional W α,λ v=vα ≡ 0 and hence also the trivial identity On the other hand, on account of the fact that −∆ α is a real operator, and so too is therefore L 1 2 , . By means of the latter two identities, (e) now gives The integral in (f) can be re-interpreted as a duality product since, owing to (3.8), and obtained from the latter by polarisation), so that (f) reads also In other words, the anti-linear maps . Now, (g) is precisely (4.1) meant as an identity in H −1 α (R 2 ).
In preparation for that, let us concisely review the abstract solution theory in the energy space for a class of semi-linear Schrödinger equations. The result imported in Proposition 5.1 below is based on a suitable generalisation, proposed in [49], of the Cazenave existence method [21,Section 3.3].
Let S be a lower semi-bounded self-adjoint operator on a Hilbert space X, with greatest lower bound m(S). The associated form domain is the energy space a Hilbert space whose scalar product is induced by the norm X * S shall denote, as customary, the topological dual of X S , thus itself equipped with a natural Hilbert space structure, in particular with Hilbert norm · X * S . For relevant functions g : X S → X * S let us consider the solution theory in X S for the semi-linear equation .
(We write g(ψ) instead of gψ to emphasise that g is not necessarily linear.) To this aim, one poses the following assumptions on g.
(G1) g is Lipschitz continuous on bounded sets of X S . (G2) g = G ′ for some G ∈ C 1 (X S , R), understanding G ′ as computed with respect to the real Hilbert space duality structure of X S and X * S , that is, ∀ψ ∈ X S ∀ε > 0 ∃ δ ψ,ε > 0 such that where ·, · X * S ,XS denotes the (complex-valued) duality pairing between X * S and X S , with the convention of being linear in the first entry and anti-linear in the second.
Proposition 5.1. Let S be a lower semi-bounded self-adjoint operator on the Hilbert space X, and let the function g : X S → X * S satisfy conditions (G1)-(G6) above. Then for any ψ 0 ∈ X S there exist a maximal time T max ≡ T max (ψ 0 ) ∈ (0, +∞] and a unique (maximal) solution ψ ∈ C([0, T max ), X S ) ∩ C 1 ([0, T max ), X * S ) to the initial value problem (5.2). Moreover: Specialising the above abstract theory to our current setting, we can establish Proposition 5.2 below, that represents the main result of this Section and will be proved in the following. In doing so, we mirror the analogous approach recently followed in [33], where the investigated two-dimensional, singular-perturbed NLS had a pure-power non-linearity, instead of our convolutive non-linearity.
For the proof of Proposition 5.2 let us first verify, by means of the inhomogeneous Strichartz estimates (2.20), the uniqueness of local weak solutions to (5.3).
, and let w satisfy (1.13). If Proof. The Duhamel formula yields, for every t ∈ (0, T ) and almost all x ∈ R 2 , and set also X T := L s1 T L r ′ 2 (with reference to the notation declared at the end of Section 1). Recall that p 1 , p 2 ∈ [1, ∞) parametrise the assumption w ∈ L p1 (R 2 ) + L p2 (R 2 ). For concreteness of presentation, let (s, r) denote for a moment either (s 1 , r 1 ), (s 2 , r 2 ), and similarly, let p stand temporarily for p 1 or p 2 . For generic T 0 ∈ (0, T ) we deduce from the above identity for ψ 1 − ψ 2 that having used the Strichartz estimate (2.20) in the first step, estimate (3.6) in space in the second, and Hölder inequality in time in the third. Observe that if (s, r) = (∞, 2), then σ = 1 and b = 2p p−1 ∈ (2, ∞) (this is the case where p > 1), and if instead (s, r) = (4, 4), then σ = 7 8 and b = 4. In either case, b is an admissible spatial index for the embedding (2.9), which gives ψ j L ∞ We can therefore conclude , for increasing n ∈ N 0 , and iterating the above reasoning a sufficient finite number of times, we finally conclude ψ 1 = ψ 2 for a.e. (t, x) ∈ [0, T ) × R 2 .
Proof of Proposition 5.2. This is an application of Proposition 5.1 with . As condition (G6) is ensured in this case by Lemma 5.4, the rest of the proof consists of checking the validity of (G1)-(G5).

Enhanced local solution theory in the mass sub-critical regime
We detour along this Section from the main line of the proof of Theorem 1.3, in order to examine more closely the mass sub-critical regime p > 1, which includes in particular the Riesz convolution potential w(x) := |x| −η , η ∈ (0, 2). We find that the solution map for the initial value problem (5.3) is locally uniformly continuous. More precisely, we obtain the following result. Proposition 6.1. Let α ∈ R and θ = ±1. Assume that w satisfy (1.13) for p := min{p 1 , p 2 } > 1 (i.e., a mass sub-critical non-linearity). Then for every M > 0 there exists T ≡ T (M ) > 0 such that the following holds: for every initial We start by collecting some useful properties of the Green function of the Laplace operator, which supplement those in Lemma 2.1. To this aim, recall that each Bessel functions K ν , ν > 0 [1, Chapter 9] is exponentially decreasing in its argument and satisfies the asymptotics [1, Eq. (9.6.9)] Lemma 6.2. For any ω > 0 one has Proof. Using the identity F (ω − ∆) 2 , together with [13, Eq. (4.1) and (4.6)], we obtain Moreover, using (2.1) and the identity K ′ 0 = −K 1 [1, Eq. (9.6.27)], we get Identities (*) and (**), together with the asymptotics (6.1) and the fact that K 1 and K 1 2 are exponentially decreasing, yield for every r ∈ [1, 2) and ε > 0, which proves (6.2)-(6.3).

Global well-posedness in the energy space
We complete in this Section the proof of Theorem 1.3 by extending the local analysis of Section 5 globally in time. Let us stress here the different reasoning needed depending on whether the non-linearity is focusing or defocusing.  (having applied (1.8) in the first step and Proposition 5.2(ii) in the third, and having used the positivity of θ in the second), whence T max = +∞ on account of Proposition 5.2(i), and sup t>0 ψ(t, ·) H 1 α < +∞. The next case to consider is the focusing (θ = −1) and mass sub-critical (p = min{p 1 , p 2 } > 1) regime. Now one has ψ(t, ·) 2 The only t-dependent summand in the r.h.s. above is controlled, by means of (3.9) and again the mass conservation, so that ψ(t, ·) 2 H 1 α α,p 1 + ψ(t, ·) Since 2 p < 2, the above estimate prevents ψ(t, ·) H 1 α to blow up in finite time, whence again T max = +∞ owing to Proposition 5.2(i), and sup t>0 ψ(t, ·) H 1 α < +∞.