On stability of rotational 2D binary Bose-Einstein condensates

We consider a two-dimensional nonlinear Schr{\"o}dinger equation proposed in Physics to model rotational binary Bose-Einstein condensates. The nonlinearity is a logarithmic modification of the usual cubic nonlinearity. The presence of both the external confining potential and rotating frame makes it difficult to apply standard techniques to directly construct ground states, as we explain in an appendix. The goal of the present paper is to analyze the orbital stability of the set of energy minimizers under mass constraint, according to the relative strength of the confining potential compared to the angular frequency. The main novelty concerns the critical case (lowest Landau Level) where these two effects compensate exactly, and orbital stability is established by using techniques related to magnetic Schr{\"o}dinger operators.


Introduction
The formation of self-bound droplets is a well-known macroscopic phenomenon.Recent experiments with ultracold quantum gases of bosonic atoms revealed a novel type of quantum liquid: dilute self-bound Bose-Einstein condensate (BEC) having orders of magnitude lower density than air (see e.g.[25,35,19,20] for Bose gases of dysprosium and [36,12] for binary Bose gases of potassium).Since these droplets form out of a BEC, there is good reason to assume that they have superfluid properties.One remarkable feature of a superfluid is its response to rotation, in particular the occurrence of quantized vortices (see [1] for a broad introduction to these phenomena).In [37], binary BEC droplets carrying angular momentum were considered.Using weak first-order corrections to the mean field energy, often referred to as the Lee-Huang-Yang correction [27], a binary BEC droplet with angular momentum is well described by the wave function ψ : R + × R 2 → C whose evolution is governed by the Gross-Pitaevskii equation (GPE) with angular momentum where the scaling invariances have been used to bring the equation into its dimensionless form.Here the external potential V is of the form where γ > 0 is the harmonic trap frequency and V 0 0 is the amplitude of the Gaussian.The parameter K 3 0 is the rate of three-body losses.The angular momentum operator L z is of the form and Ω > 0 is the rotational speed.The fact that the constants in the harmonic trap and the Gaussian part of the potential are equal stems from [37], but is not crucial in our analysis, so we consider more generally , γ, γ 0 > 0. (1.4) The main purpose of this paper is to study the existence/nonexistence and orbital stability of mass-constraint standing waves for (1.1).We consider the Cauchy problem for (1.1) with initial data ψ 0 ∈ Σ, where is equipped with the norm Due to the presence of the harmonic potential, this space is rather natural (see e.g.[14]).In the case K 3 = 0, there are three physical quantities which are formally conserved along the flow of (1.1) ψ(t, x)L z ψ(t, x)dx = L(ψ 0 ), (Angular momentum) Here we note that the angular momentum is real-valued, but has no definite sign.Similarly, the term involving the natural logarithm also has no definite sign.In the case K 3 = 0, solutions to (1.1) formally satisfy 1 2 This shows that for K 3 = 0, the equation is irreversible.This is the reason why, since K 3 0, we consider only positive time in the present paper.In the case K 3 = 0, the equation is reversible, and ψ(−t, x) solves (1.1): considering the case t 0 suffices to describes the dynamics for all time.
When γ = V 0 = K 3 = Ω = 0, the equation (1.1) was recently studied in [16].More precisely, the global well-posedness for H 1 data, the existence and uniqueness of positive ground state solutions for (1.1) were shown there, along with the orbital stability of prescribed mass standing waves.
Our first result is the following global existence for (1.1).
Theorem 1.1 (Global well-posedness).Let γ, γ 0 > 0, Ω > 0, V 0 0, K 3 0, and ψ 0 ∈ Σ.Then there exists a unique global-in-time solution to (1.1) satisfying ψ ∈ C(R + ; Σ) ∩ L 3 loc (R + ; L 6 (R 2 )).• If K 3 = 0, the conservation laws of mass, angular momentum, and energy hold.• If K 3 > 0, the solution asymptotically vanishes in the sense that The Cauchy problem is addressed by resuming the approach from [3].In passing, we fix a small flaw present in this paper regarding dispersive estimates.The asymptotic extinction (1.6) is then established like in [2], thanks to a suitable uniform bound which makes it possible to control the L 6 -norm in (1.5) from below in terms of the L 2 -norm.
Remark 1.2.The potential V in (1.4) is radially symmetric, since it is the model given in [37].We will see in the proof of Theorem 1.1 that the result still holds true in the more general case of smooth potentials which are at most quadratic (and thus need not be radial); see Remarks 2.2 and 2.3.
In the rest of the introduction, we are interested in the absence of three-body losses, i.e.K 3 = 0.In this case, (1.1) admits standing waves, i.e. solutions of the form where φ solves (1.8) Note that in the case K 3 > 0, there is no such solution in view of the asymptotic extinction (1.6).
The existence of standing waves for (1.8) can be achieved by several ways.The first way is to minimize the energy functional with prescribed mass constraint, i.e. f 2 L 2 = ρ > 0, an strategy which is often adopted in Physics.In this case, the parameter ω in (1.8) appears as a Lagrange multiplier associated to the minimization problem.Another way is to look for critical points of the action functional with ω being given and fixed.However, this approach seems difficult to apply in the present context.More precisely, (1.8) has two features which make it difficult to characterize the range of ω's allowed to find a non-trivial solution to (1.8).The presence of the external potential V and the rotation L z introduces an x-dependence which makes it impossible to invoke the results from [9] (the 2D counterpart of [10]), or even adapt easily the proof.On the other hand, the fact that the nonlinearity is not homogeneous in φ makes it impossible to reproduce the arguments from [34] (see also [22] in the case of a harmonic potential).In an appendix, we collect some information regarding the possible range for ω in the radial case, where the rotating term is absent from (1.8), and explain in more details why minimizing the action seems difficult here.We therefore consider the following minimization problem: for ρ > 0, Our next result concerns the existence and stability of prescribed mass standing waves for (1.8) in the case of low rotational speed.
Next we consider the critical rotational speed Ω = γ (lowest Landau level, see e.g.[1] and references therein), which constitutes the main novelty of this paper.In this case, the energy functional can be rewritten as where ). Thanks to this observation, it is convenient to consider the above functional on the magnetic Sobolev space endowed with the norm Note that the energy functional (1.11) is well-defined on H 1 A (R 2 ) by using (1.10) and the magnetic Gagliardo-Nirenberg inequality (see e.g., [18]): for 2 < r < ∞, where the last inequality follows from To show the existence and stability of standing waves to (1.1) in the critical rotational case, it is convenient to consider separately two cases: V 0 = 0 and V 0 > 0.
When V 0 = 0, we denote where and the superscript 0 is here to emphasize the assumption V 0 = 0. Let us first introduce the cubic ground state, that is, the unique positive, radially symmetric solution to By making use of a variant of the celebrated concentration-compactness principle of Lions adapted to the magnetic Sobolev space H 1 A (R 2 ) (see Lemma 3.8), we prove the following result.
Moreover, the set of minimizers for I 0 γ (ρ) is orbitally stable under the flow of (1.1).
The assumption ρ Q 2 L 2 is probably technical, due to our argument (see after (3.18)).The proof also relies on the property I 0 γ (ρ) < 0 (see Proposition 3.5).In the case K 3 = V 0 = γ = Ω = 0, the latter condition was shown in [16] for any ρ > 0. In our setting, showing that I 0 γ (ρ) < 0 is more complicated since the scaling argument used in [16] does not work because of the presence of magnetic potential.However, by using a trial function of the form λe −b|x| 2 with suitable positive constants λ and b, we are able to show (see Lemma 3.10) that both conditions ρ Q 2 L 2 and I 0 γ (ρ) < 0 are fulfilled by some data f ∈ H 1 A (R 2 ).This also leads to a restriction on the validity of γ.We refer the reader to Subsection 3.2 for more details.
When V 0 > 0, the above-mentioned concentration-compactness argument does not work due to the lack of spatial translation of the term To overcome the difficulty, we restrict our consideration on H 1 A,rad (R 2 ) the space of radially symmetric functions of H 1 A (R 2 ).Note that this restriction has a drawback since we no longer see the effect of rotation to the equation as L z f = 0 for a radial function f .We consider where E γ (f ) is as in (1.11).We have the following result.
Remark 1.6.As mentioned above that the rotation term vanishes for radially symmetric functions, the result in Theorem 1.5 holds for any Ω ≥ 0. In particular, it gives another (radial) solution, besides the one obtained in Theorem 1.4, to (1.8).
The proof of Theorem 1.5 relies on the following compact embedding for all 2 r < ∞.This compact embedding follows from the well-known compact embedding H 1 rad (R 2 ) ֒→ L r (R 2 ) for all 2 r < ∞ and the fact that is continuous due to the diamagnetic inequality (see e.g.[28]) When the rotational speed exceeds the critical value, i.e.Ω > γ, we have the following nonexistence of minimizers for the constrained variational problem I Ω (ρ).
The proof of Theorem 1.7 is based on an idea of Bao, Wang, and Markowich [7, Section 3.2], using the central vortex state with winding number m, namely where (r, θ) are the polar coordinates in R 2 .Physically, when the angular velocity of rotation exceeds the trapping frequency, the harmonic potential cannot provide enough necessary centripetal force that counteracts the centrifugal force caused by the rotation, and the gas may fly apart.In the classical rotating BEC with purely power-type nonlinearity, the nonexistence of prescribed mass standing waves was proved by Bao, Wang, and Markowich [7].This work is organized as follows.In Section 2, we study the local well-posedness for (1.1) and prove the global existence given in Theorem 1.1.In Section 3, we investigate the existence/nonexistence and the orbital stability of prescribed mass standing waves for (1.1) given in Theorems 1.3, 1.4, 1.5, and 1.7.Finally, we give some information on stationary solutions of (1.8) in the radial case in Appendix A, and characterize prescribed mass minimizers in Appendix B.

Cauchy problem
In the case K 3 = 0, Theorem 1.1 is reminiscent of the results from [3] for the local and global well-posedness.We present an alternative proof, to show that local solutions are actually global, which simplifies the approach, and yields better bounds.The asymptotic extinction in the case K 3 > 0 is obtained by adapting the arguments from [4] based on the introduction of a suitable pseudo-energy (see also [2] for generalizations).
where here and in the following, we denote The notation U (•) stands for the linear propagator, that is, U (t)ψ 0 = ψ lin (t) is the solution to As noticed in [3], Dx) , where enters into the general framework of [26], where the fundamental solution, that is, the kernel associated to U (•), is constructed.However, this does not yield directly local in time dispersive properties, since unlike in [21], the solution to for H = H(t, x, ξ) smooth, real-valued and at most quadratic in (x, ξ) (locally in time), is represented as e iφ(t,x,ξ) a(t, x, ξ) φ(ξ)dξ, 1 We do not include here the semi-classical parameter present in [21,26].
that is, an oscillatory integral involving the Fourier transform of ϕ, and not ϕ directly.Consequently, dispersive L 1 − L ∞ estimates cannot be inferred in general, as shown by the trivial case H = 0: it enters the framework of [26], the above representation holds with a = constant (whose value depends on the definition of the Fourier transform), but no Strichartz estimate is available, of course.
In the present case, local dispersive estimates are available though, as we show by relying on the isotropy of V and L z ; this corresponds to the trick presented in [3] in the 3D case.This alternative approach not only yields dispersive estimates, but also makes it possible to study the Cauchy problem as if the rotation term was absent.Define the function ϕ by (2.3) We check that since V depends only on |x| 2 (and is therefore invariant under rotation), ψ solves (1.1) if and only if that is, (1.1) with Ω = 0. Now for the perturbed harmonic oscillator H = − 1 2 ∆+V , local in time dispersive estimates are available: from [21], there exists Note that δ is necessarily finite, as H possesses eigenvalues (as a consequence from e.g., [33,Theorem XIII.67]).This implies local in time Strichartz estimates (see e.g., [17]) and so, as long as bounded time intervals only are involved, (2.1) can be studied like in the case where U (t) = e it∆ .Note that since (2.3) preserves the Lebesgue norms, we infer for all Ω ∈ R, with δ and an implicit multiplicative constant independent of Ω.
In the case K 3 = 0, the analysis meets essentially the one presented in [16].In the general case We then obtain: Then there exist T > 0 and a unique solution In the case K 3 = 0, we also have the conservation laws of mass, angular momentum, and energy.Sketch of the proof.We present the elements of the proof which require a little modification due to the presence of the damping term (K 3 > 0).
We recall that the standard blow-up alternative involves the condition In view of (1.5), the L 2 -norm of ψ remains bounded for positive time, and thus, so does the L 2 -norm of ϕ.On the other hand, we compute Using Cauchy-Schwarz and Young's inequalities, we infer that Therefore, if ∇ϕ remains bounded in L 2 , then so does the Σ-norm, and the blow-up criterion can be reduced to the one stated in the lemma.
In the case K 3 = 0, the proof of conservation laws follows, for instance, from [31].
Remark 2.2 (Non-radial potential).The case of a non-radial potential can be treated along essentially the same lines.Indeed, for a potential V ∈ C ∞ (R 2 ; R), which is at most quadratic in the sense of [21], that is, , the function ϕ, given by (2.3), solves ).The time dependent potential Ṽ is smooth in (t, x), and at most quadratic in space (in the same sense as above), uniformly in time.It follows from [21] that (2.5) remains valid with e −itH replaced by the evolution group U (s + t, s) associated to H(t) = − 1 2 ∆ + Ṽ (t, x), hence (2.6) holds in this case too.The rest of the proof of Lemma 2.1 can be repeated exactly then.

2.2.
Global well-posedness.In the case K 3 = 0, the following energy is independent of time, as well as the mass of ϕ.The positive part of the energy satisfies where we have used the easy bound ln a a for a 1.Using the conservation of mass, this yields and thus global existence (in the past as well, since for K 3 = 0, the equation is reversible), since ϕ Σ E + (ϕ).Note that even the L 2 -norm of ϕ is controlled by E + , in view of the uncertainty principle (see e.g., [32, § 3.2]) In particular, there exists C > 0 such that (2.9) In the case K 3 > 0, following the strategy presented in [4, Section 3.1], we introduce a pseudo-energy, for k > 0, Adapting slightly the computations from [2], we find: Again, following the computations from [4, Section 3.1], we introduce the polar factor related to ϕ, and the above time derivative can be rewritten as Picking 0 < k < K 3 /6, the above relation implies Since the change of unknown (2.3) preserves the Lebesgue norms, (1.5) yields and we conclude like in the case K 3 = 0.In particular, (2.9) holds.
Remark 2.3 (Non-radial potential, continued).For a general potential V like in Remark 2.2, the natural energy associated to ϕ becomes and even for K 3 = 0, it is not constant in time if V is not radial (∂ t Ṽ ≡ 0).However, following either the virial computation from [13] or the pseudo-energy argument from [15], we can easily adapt the above argument and infer that the global well-posedness in Theorem 1.1 remains valid.
L 6 , where we have used (2.9).This, together with (1.5), yields for some uniform C > 0. Recalling that the (nonnegative) solution of the ODE ẏ + Cy 5 = 0 with y(0 Again, this conclusion remains true when V satisfies the assumptions of Remark 2.2 and is not necessarily radial.

Orbital stability
In this section, we study the existence/nonexistence and stability of constraint mass standing waves associated to (1.8).Recall that K 3 = 0 throughout this section.We will consider separately three cases: low rotational speed (0 < Ω < γ), critical rotational speed (Ω = γ), and high rotational speed (Ω > γ).
3.1.Low rotational speed.In this subsection, we consider the low rotational speed 0 < Ω < γ.Let us start with the following observation, mimicking the proof of [5, Proposition 2.1] (see also [8,Theorem 7.8]): Proof.We first observe that by Cauchy-Schwarz and Young inequalities, we have for any δ > 0, Using (3.2) with δ = 1 2 , we have , where we have used (2.8) to get the third line.On the other hand, by (3.2), we have for any δ > 0, We choose δ > 0 so that The proof is complete.
Proof of Theorem 1.3.The proof is divided into two steps.
Step 1.We show the existence of minimizers for I Ω (ρ).Let ρ > 0 and f ∈ Σ satisfy f 2 L 2 = ρ.We first show that I Ω (ρ) is well-defined.We have where B(f ) is as in (3.3).Here we have used the fact that ln(1/λ) < 1/λ for 0 < λ < 1.Using (3.1), we see that Next let (f n ) n 1 be a minimizing sequence for I Ω (ρ).By (3.4) and (3.1), we see that (f n ) n 1 is a bounded sequence in Σ.Thus there exists φ ∈ Σ such that (up to a subsequence) f n → φ weakly in Σ and strongly in L r (R 2 ) for all 2 r < ∞ (see e.g., [33,Theorem XIII.67] for the compactness of the embedding Σ ֒→ L r (R 2 )).It follows that By the strong convergence, Hölder's inequality, and the fact that we infer that Moreover, by the lower semicontinuity of the weak convergence and (3.1), we have which together with (3.5) show that φ is a minimizer for I Ω (ρ).In view of Lemma 3.1, we also have that (up to a subsequence) f n converges to φ strongly in Σ.
Step 2. We show the orbital stability.We argue by contradiction.Suppose that G Ω (ρ) is not orbitally stable.There exist ǫ 0 > 0 and φ 0 ∈ G Ω (ρ) and a sequence u 0,n ∈ Σ satisfying and a sequence of time where u n is the solution to (1.1) with initial data u n (0) = u 0,n .Since φ 0 ∈ G Ω (ρ), we have E Ω (φ 0 ) = I Ω (ρ).By (3.7), (3.1) and Sobolev embedding, we have By the conservation of mass and energy, we get This shows that (u n (t n )) n 1 is a minimizing sequence for I Ω (ρ).By Step 1, we see that up to a subsequence, u n (t n ) → φ strongly in Σ for some φ ∈ G Ω (ρ) which contradicts (3.8).The proof is now complete.
3.2.Critical rotational speed.In this subsection, we study the existence and stability of standing waves related to (1.8) with a critical rotational speed Ω = γ (lowest Landau level).To this end, we first collect some basic properties of the magnetic Sobolev space H 1 A (R 2 ) (see (1.12)) in the following lemma.Lemma 3.2 ( [18,28]).We have: (3.9) Lemma 3.3.The best constant in (3.10) is the same as the optimal constant in Gagliardo-Nirenberg inequality for the case A = 0.In particular, when r = 4, the best constant in (3.10) can be taken as , where Q is the cubic (positive, radial) ground state given by (1.15).Moreover, the equality in (3.10) cannot be attained.
Proof.We recall the standard Gagliardo-Nirenberg inequality where C GN r stands for the best constant in the "regular" Gagliardo-Nirenberg inequality.This best constant was first characterized in [38], in terms of a suitable ground state solution of some elliptic equation.In particular, C GN 4 is given by , where Q is the (only) positive, radially symmetric solution to (1.15) (recall that unlike in [38], there is a 1  2 factor in front of the Laplacian in (1.15), hence a slightly different formula).As a consequence of the diamagnetic inequality (3.9), we infer . This shows that (3.10) holds and C r C GN r .To see C GN r C r , we follow the argument of [11,Proposition 3.1].Note that the argument given [11] does not hold in two dimensions.Denote A σ (x) := σA(σx) with σ > 0. By (3.10), we have for any smooth compactly supported function f , L 2 , ∀σ > 0. On the other hand, by the triangle inequality, we have We estimate where R > 0 is such that supp(f ) ⊂ {x ∈ R 2 : |x| R}.This shows that Finally we show that the equality in (3.10) cannot be achieved.Assume by contradiction that (3.10) is attained by some function φ.By (3.9) and (3.11), we have . This together with the following estimate of [28,Theorem 7.21]: In particular, . This is a contradiction since B 12 (x) = 2.The proof is complete.
. We have the following conditional result on the existence and stability of prescribed mass standing waves to (1.1).
ρ is orbitally stable under the flow of (1.1) in the sense that for any ǫ > 0, there exists δ > 0 such that for any initial data the corresponding solution to (1.1) exists globally in time and satisfies Remark 3.6.In the case A ≡ 0 and γ = 0, it was shown in [16] that I(ρ) := I 0 0 (ρ) < 0 for any ρ > 0. This is done by using the scaling f λ (x) := λf (λx) and taking λ > 0 sufficiently small.In our case, showing I 0 γ (ρ) < 0 is more complicated.The above scaling argument does not work due to the presence of magnetic potential A.
Before giving the proof of Proposition 3.5, let us recall the following version of concentration-compactness lemma.Lemma 3.8 (Concentration-compactness lemma [29,30]).
1 with some fixed constant ρ > 0. Then there exists a subsequencestill denoted by (f n ) n 1 -satisfying one of the following three possibilities: • Dichotomy.There exist a ∈ (0, ρ) and sequences

13)
• Compactness.There exists a sequence (y n ) n 1 ⊂ R 2 such that for all ǫ > 0, there exists R(ǫ) > 0 such that for all k 1, The proof of this result is very similar to that of [29,30], see also [17, Proposition 1.7.6].The only difference is the last inequality in (3.13) which is proved by using the fact that Remark 3.9.
• If vanishing occurs, then we infer that f n → 0 strongly in L r (R 2 ) for all 2 < r < ∞.Indeed, by the magnetic inequality (3.9), we see that |f n | is a bounded sequence in H 1 (R 2 ).The result follows by applying [30 r < ∞ up to a translation and change of gauge, i.e. there exists (y n ) n 1 ⊂ R 2 such that up to subsequence ).In fact, denote fn (x) := e iA(yn)•x f n (x + y n ).
We see that It follows that ( fn ) n 1 is a bounded sequence in H 1 A (R 2 ) satisfying for all ǫ > 0, there exists R(ǫ) > 0 such that for all n 1, By the standard diagonalization argument and the compact embedding H 1 A (R 2 ) ֒→ L r (B(0, R)) for all R > 0 and all 2 r < ∞, we show that fn → φ strongly in L 2 (R 2 ), hence strongly in L r (R 2 ) for all 2 r < ∞ by Sobolev embedding.
Proof of Proposition 3.5.The proof is divided into three steps.
No vanishing.Suppose that vanishing occurs.Passing to a subsequence, we infer that f n → 0 strongly in L r (R 2 ) for all 2 < r < ∞.Thanks to (1.10), we infer that which contradicts the assumption I 0 γ (ρ) < 0. No dichotomy.If dichotomy occurs, then there exist a ∈ (0, ρ) and sequences ) such that (3.13) holds.To rule out the dichotomy, we first claim that there exists δ > 0 such that lim inf for n sufficiently large.The claim (3.15) follows from the Hölder inequality f 3 We next claim that lim inf To see this, we consider K(z) = z 4 ln(z 2 ) for z > 0. Using Taylor expansion and the fact that for any ǫ > 0, we infer that Applying the above estimate with ǫ = 1, N = 3 and e n := f n − g n − h n , we see that Using (1.10), Hölder inequality and the fact that e n → 0 strongly in L r (R 2 ) for all 2 r < ∞, we have Similarly, we can prove that This shows that Using this and (3.13), we show (3.16).Now, let λ > 0. We have which implies We see that for n sufficiently large, Here we have used the magnetic Gagliardo-Nirenberg inequality (3.10) and the fact λ n → ρ a > 1 as n → ∞.A similar estimate goes for E 0 γ (h n ).Thus, we get Passing n → ∞ and using (3.16), we obtain where Here K 1 , K 2 > 0 as soon as C 4 ρ 1, which follows from 0 < ρ Q 2 L 2 , see Lemma 3.3.Using (3.15) and (3.17), we infer that I 0 γ (ρ) > 0 which is a contradiction.
Compactness.Therefore, compactness must occur.In this case, there exist φ ∈ H 1 A (R 2 ) and (y n ) n 1 ⊂ R 2 such that up to a subsequence, fn → φ strongly in L r (R 2 ) for all 2 r < ∞, where fn (x) := e iA(yn)•x f n (x + y n ).It follows that We also have Thus, we get This shows that φ is a minimizer for I 0 γ (ρ).We also have that fn → φ strongly in Step 3. The orbital stability of G 0 γ (ρ) follows by the same argument as in Theorem 1.3.We thus omit the details.The proof is now complete.
In the following lemma, we show that the conditions 0 < ρ Q 2 L 2 and I 0 γ (ρ) < 0 are satisfied for some data Proof.We look for a function To this end, we consider f b (x) = e −b|x| 2 for some b > 0 to be chosen later.A direct computation shows We also have In addition, we have Let λ > 0 to be chosen later.We see that To make λf b In view of (3.12), this is granted by the property λ 2 2b.Consider We rewrite The condition E 0 γ (λf b ) < 0 is now reduced to finding θ ∈ (0, 2b] so that G(θ) < 0. We have If 2b 1, we see that G attains its minimum at θ = 1, however G(1) = 4b 2 +γ 2 −b > 0 for 2b 1.So, we need 2b < 1.In this case, G is strictly decreasing on (0, 2b]. To find θ ∈ (0, 2b] so that G(θ) < 0, a necessary condition is We see that on 0, 1  2 , H attains its minimum at b , then we have G (2b 0 ) < 0. Therefore, by choosing λ > 0 so that λ 2 2b 0 , we show the existence of a function f ∈ H 1 A (R 2 ) satisfying (3.19).The proof is complete.
Proof of Theorem 1.4.It follows directly from Proposition 3.5 and Lemma 3.10.
In the case V 0 > 0, the concentration-compactness argument presented above does not work due to the lack of spatial translation in the new potential term More precisely, the sequence (y n ) n 1 ⊂ R 2 , which appeared in the compactness, may tend to infinity, and then To overcome this difficulty, we restrict our consideration on H 1 A,rad (R 2 ) the space of radially symmetric functions of H 1 A (R 2 ).Note that this restriction has the drawback that we no longer see the effect of rotation to the equation since L z f = 0 for a radial functions f .Proof of Theorem 1.5.We proceed in two steps.
Step 1.We show that there exists a minimizer for I γ,rad (ρ).Let ρ > 0 and We have This shows that ) is compact for all 2 r < ∞, there exists φ ∈ H 1 A,rad (R 2 ) such that, up to a subsequence, f n → φ weakly in H 1 A,rad (R 2 ) and strongly in L r (R 2 ) for all 2 r < ∞.By the strong convergence and (3.6), we see that On the other hand, by the weak continuity of the potential energy (see e.g., [28,Theorem 11.4]), we have Moreover, the lower semicontinuity of the weak convergence implies Thus we get which together with implies that φ is a minimizer for I γ,rad (ρ).In addition, we have that (up to a subsequence), Step 2. The orbital stability follows from the same argument as in the proof of Theorem 1.3.We omit the details.
It is useful to recall the following formulas: We have Thus we get The condition We compute It follows that Thus we have Thanks to (3.21) and (3.22), we have which, together with (3.21) and (3.22), implies that Moreover, since the rotation operator can be rewritten as L z = −i∂ θ , we have 3 , which tends to zero as m → ∞ for each ρ > 0 fixed.By interpolation between L 2 and L 6 , we infer that Finally we have Therefore, for any ρ > 0, there is no minimizer for I Ω (ρ).

Appendix A. Ground states
In this appendix, we collect some information on stationary solutions in the radial case, where the rotation terms is absent, so (1.8) becomes Recall that We introduce the minimizing problem Denoting by we see that ω V0 is simply the bottom of the spectrum of H.If V 0 = 0, then it is well-known that ω 0 is attained by the Gaussian , and ω 0 = γ.
Lemma A.1.Let V 0 0. Then ω V0 is attained and γ = ω 0 ω V0 γ + γ γ+γ0 V 0 .Proof.Since V 0 0, it is obvious that ω V0 ω 0 .Also, hence ω V0 γ + γ γ+γ0 V 0 .It remains to show that ω V0 is attained.Let (f n ) n 1 be a minimizing sequence for ω V0 .It follows that (f n ) n 1 is a bounded sequence in Σ.Thus, there exists ϕ ∈ Σ such that (up to a subsequence) f n → ϕ weakly in Σ and strongly in L r (R 2 ) for all 2 r < ∞.We infer that By the lower continuity of the weak convergence, we have We also have from the weak continuity of the potential energy (see e.g., [28,Theorem 11.4]) that hence ϕ is a minimizer for ω V0 .The proof is complete.
Consider now the Pohozaev identities related to (A.1).
Lemma A.2. Let φ ∈ Σ be a solution to (A.1).Then the following identities hold In particular, since V 0 0, if ω Proof.The identity (A.4) follows by multiplying both sides of (A.1) with φ and taking the integration over R 2 .By multiplying both sides of (A.1) with x • ∇ φ, integrating over R 2 , taking the real part and using the fact that Re we get (A.5).Finally, (A.6) follows directly from (A.4) and (A.5).
In view of the formula In particular, when V 0 = 0, the above conditions become ω < 1 2 √ e and ω + γ 1 e .
According to the value of γ, either the first condition or the second one is the more stringent.
We now explain in more details why finding directly a solution to (A.1) (that is, not minimizing the energy under a mass constraint, in which case ω corresponds to the -unknown -Lagrange multiplier) seems intricate.In order to simplify the computations, we go back to the initial case (1.2), that is, we assume γ 0 = γ.To find ground states related to (A.1), a standard way is to consider the following minimization problem and then show that minimizers for m ω are indeed solutions to (A.1), where We have the following sufficient condition that ensures the minimizing problem (A.9) is well-defined.
For λ > 0 to be chosen later, we have Using (A.7), we see that the last term takes its maximal negative value Optimizing the sum of the first and third terms yields b = γ 2 , hence By the continuity argument, there exists f ∈ Σ\{0} such that K ω (f ) = 0.
Remark A.5.The condition (A.12) is stronger than (A.8) since V 0 0 and ω V0 γ + V0 2 .By the above remark, from now on, we consider the following condition on ω This condition is equivalent to To make the above inequality is not empty, we need Note that the left hand side belongs to 0, V0 2 as γ ω V0 γ + V0 2 .So, if V 0 < 1 √ e ≃ 0.6, then the above condition is satisfied.Note that only the values V 0 = 0 and V 0 = 0.2 are considered in [37].
To show the existence of minimizers for m ω , we need an estimate showing the boundedness in Σ of minimizing sequences for m ω .However due to the interplay of various nonlinear terms, it is not clear to us how to prove this at the moment.Below we some estimates which may be helpful for future investigation.
Remark A.6.Let ω be as in (A.13).Let f ∈ Σ\{0} be such that K ω (f ) = 0. Then there exist positive constants Indeed, since K ω (f ) = 0, we have By the definition of ω V0 , we see that By the Gagliardo-Nirenberg inequality, we obtain We have the following useful estimates: and where Q is the unique positive radial solution to (1.15) satisfying (3.12).
In the following remark, we point out another difficulty in finding ground states related to (A.1).More precisely, suppose that there is a minimizer φ for m ω , we are not able to show that φ is a solution to (A.1).We want to show that λ = 0 so that φ is a solution to (A.1).To see this, we first multiply both sides of (A. 16) with φ and integrate over R 2 to get Suppose that λ = 0, then we must have In particular, as K ω (φ) = 0, we have From this and (A.17However, it is not clear that λ = 0, and it is not possible to conclude.

Appendix B. Characterization of prescribed mass minimizers
In this section, we will characterize the orbit of prescribed mass standing waves obtained in Section 3, in the absence of the rotational effect, following the strategy from [24].Throughout this section, we assume that Ω = 0 and K 3 = 0. Denote For f ∈ Σ R , we define Let z = (f, g) ≃ f + ig.We observe that For z ∈ Σ, we consider Here | • | L 2 and • L 2 denote the norms of L 2 (R 2 , R) and L 2 (R 2 , C), respectively.For c > 0, we consider the following minimizing problems: Proof.The proof is essentially given in [24].For the reader's convenience, we provide some details.
(1) Let z = (f, g) ∈ S c be such that E 0 (z) = I 0,c and set ϕ := |z| = f 2 + g 2 .It follows that ϕ ∈ S R c and we have (see [24,Proposition 2.2.]) for i = 1, 2, We see that On the other hand, as E 0 (z) = I 0,c , there exists µ ∈ C such that for any ξ ∈ Σ, Putting z = ξ and using regularity theory, we can deduce that f and g belong to C 1 (R 2 ).Now suppose that f ≡ 0. Denote The continuity of f implies that δ f is closed.Let us now prove that it is also an open set of R 2 .Pick a point x 0 ∈ δ f .Since ϕ(x 0 ) > 0, there exists a ball B(x 0 , ρ) centered at x 0 with radius ρ > 0 such that g(x) = 0 for all x ∈ B(x 0 , ρ).Observe that for each x ∈ B(x , ρ) and i = 1, 2, (f From this and (B.3), we have Hence ∇ f g = 0 on B(x 0 , ρ).This implies that f g = C on B(x 0 , ρ) for some constant C > 0. As f (x 0 ) = 0, we infer that C = 0. Hence f (x) = 0 for all x ∈ B(x 0 , ρ) or B(x 0 , ρ) ⊂ δ f which tells us that δ f is an open set.
Second case: If g(x) = 0 on R 2 , then for i = 1, 2, we have from (B.3) that

2. 1 .
Local well-posedness.Local well-posedness follows from a fixed point argument on Duhamel's formula

2. 3 .
Large time extinction in the presence of loss.In view of [2, Lemma 4.1],