On an intercritical log-modified nonlinear Schrodinger equation in two spatial dimensions
aa r X i v : . [ m a t h . A P ] J un ON AN INTERCRITICAL LOG-MODIFIED NONLINEARSCHR ¨ODINGER EQUATION IN TWO SPATIAL DIMENSIONS
R´EMI CARLES AND CHRISTOF SPARBER
Abstract.
We consider a dispersive equation of Schr¨odinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation issupposed to be an effective model for stable two dimensional quantum dropletswith LHY correction. Mathematically, it is seen to be mass supercritical andenergy subcritical with a sign-indefinite nonlinearity. For the correspondinginitial value problem, we prove global in-time existence of strong solutions inthe energy space. Furthermore, we prove the existence and uniqueness (up tosymmetries) of nonlinear ground states and the orbital stability of the set ofenergy minimizers. We also show that for the corresponding model in 1D astronger stability result is available. Introduction
In this paper we consider the Cauchy problem for the following log-modifiednonlinear Schr¨odinger equation (NLS) on R :(1.1) i∂ t u + 12 ∆ u = λu | u | ln | u | , x ∈ R , λ > ,u (0 , x ) = u ∈ H ( R ) . This model is discussed in the physics literature (cf. [26, 31, 33]) as an effectivemean-field description of ultra-dilute quantum fluids in two spatial dimensions. Thelogarithmic factor thereby stems from the LHY-correction (after Lee-Huang-Yang),a series expansion in the mean particle density of Bose-Einstein condensates withorigins in the work of Bogolubov (see, e.g., [22, 30] for more details). It is arguedthat the LHY correction should have a stabilizing effect on an otherwise collapsingcondensate, allowing for stable soliton-like modes, which are often called quantumdroplets . Unfortunately, there are only a few results available to date concerningthe rigorous mathematical derivation of the LHY correction, the most recent being[4] concerning second order corrections to the (mean-field) bosonic ground stateenergy in three spatial dimensions. The corresponding problem in 2D, however,still remains open.Nevertheless, the NLS (1.1) has several mathematical properties which make itan intriguing model to study: It can be seen as the Hamiltonian evolution equationassociated to the following energy functional (1.2) E ( u ) := 12 k∇ u k L ( R ) + λ Z R | u | ln (cid:18) | u | √ e (cid:19) dx. Date : June 25, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Nonlinear Schr¨odinger equation, solitary waves, orbital stability.RC is supported by Rennes M´etropole through its AIS program. CS acknowledges support bythe NSF through grant no. DMS-1348092.
The latter is thus (at least formally) conserved by solutions to (1.1), as are the total mass and momentum , i.e.,(1.3) M ( u ) := Z R | u | dx, P ( u ) := Z R Im u ∇ u dx. In view of (1.2), one sees that the second term in the energy, i.e., the one stemmingfrom the nonlinearity, has no definite sign. Indeed, in terms of the usual classifi-cation of NLS (see, e.g. [8]), the nonlinearity in (1.1) is seen to be defocusing (orrepulsive) whenever the density | u | > √ e and focusing (or attractive) whenever | u | < √ e . Furthermore, it is well known that in the case of pure power-law nonlin-earities such as λ | u | p − u , solutions u to NLS obey the additional scaling symmetry u ( t, x ) u µ ( t, x ) = µ / ( p − u ( µ t, µx ) , µ > . In two spatial dimensions, this implies that the cubic case p = 3 is mass-critical ,since in this case the transformation also preserves the L ( R )-norm of u . Ithas been proved, that the corresponding Cauchy problem is globally well-posedin L ( R ) in the defocusing case, and also in the focusing case for masses belowthe one of the ground state (cf. [13, 14] for more details). Furthermore the Cauchyproblem becomes ill-posed if one tries to study it in spaces which are less regularthan L ([20]).Coming back to our model, we first note that due to the appearance of thelogarithmic factor, (1.1) does not obey any scaling symmetry. However, since forall ε >
0, we have (cid:12)(cid:12) u | u | ln | u | (cid:12)(cid:12) . | u | − ε + | u | ε , the log-modified NLS can formally be seen to be inter-critical , in two different ways:First, its nonlinearity is slightly larger than cubic, and thus mass supercritical,but still remains energy subcritical. Second, it can be understood as the sumof a slightly L -subcritical (focusing) nonlinearity and a slightly L -supercritical(defocusing) nonlinearity. It is therefore similar to the case of NLS with competingcubic-quintic power law nonlinearities, i.e.(1.4) i∂ t u + 12 ∆ u = −| u | u + | u | u, which has been studied in [21] in 3D, and, more recently, in [7, 23] in various spacedimensions.Our first main result of this work is as follows: Theorem 1.1 (Global well-posedness) . For any u ∈ H ( R ) , there exists a uniqueglobal in-time solution u ∈ C ( R ; H ( R )) ∩ C ( R ; H − ( R )) to (1.1) , depending con-tinuously on the initial data u . Furthermore the solution u obeys the conservationof mass, energy and momentum. This result can be interpreted as a rigorous expression of the stabilizing effect ofthe LHY correction in two spatial dimensions. Recall that the focusing, cubic NLSin two spatial dimensions, in general, exhibits finite-time blow-up of solutions. Theintroduction of the logarithmic factor prevents any such blow-up from happening.
Remark . Since (1.1) is a logarithmic perturbation of the L -critical case, localand global well-posedness might even hold in L ( R ), in view of the similar case ofa (smooth) logarithmic perturbation of an energy-critical wave-equation consideredin [32].Our second main result concerns the properties of solitary waves , i.e., solutionsof the form u ( t, x ) = e iωt φ ( x ) , ω ∈ R , NTERCRITICAL LOG-MODIFIED SCHR ¨ODINGER EQUATION 3 where φ solves(1.5) −
12 ∆ φ + λφ | φ | ln | φ | + ωφ = 0 , x ∈ R . Clearly, solutions to this equation can only be unique up to translations and phaseconjugation, a fact that, together with the Galilei invariance of (1.1), allows one tosubsequently construct more general solitary waves, moving with non-zero speed.In the following we shall denote the action associated to (1.5) by S ( φ ) = E ( φ ) + ωM ( φ ) . A solution φ is called a nonlinear ground state if it minimizes the action S ( φ )among all possible solutions φ of (1.5). It follows from [10, Lemma 2.3] and [5,Proposition 4] that every minimizer ϕ of the action S ( φ ) is of the form ϕ ( x ) = e iθ φ ω ( x − x ) , for some constants θ ∈ R , x ∈ R , and where φ ω is a positive least action solutionto (1.5). The existence and uniqueness of such positive ground states is the contentof our second main result. Theorem 1.3 (Existence and uniqueness of positive ground states) . Suppose thatthe frequency ω ∈ R satisfies < ω < λ √ e . Then (1.5) admits a unique solution φ ω ∈ C ( R ) which is radially symmetric andexponentially decaying as | x | → ∞ . Moreover, for all x ∈ R , it holds < φ ω ( x ) < √ z ω , for some uniquely defined parameter z ω ∈ ( e , , which satisfies z ω → as ω → + . These ground states can be physically interpreted as quantum droplets with zerovorticity. In numerical simulations, they are found to have a rather flat top withnearly constant value of the density in its interior, see [26].As a final result we shall turn to the question of orbital stability of solitary waves.To this end we first recall the following notions for constrained energy minimizers.
Definition 1.4.
For ρ >
0, denoteΓ( ρ ) = (cid:8) u ∈ H ( R ) , M ( u ) = ρ (cid:9) . Assuming that the minimization problem(1.6) u ∈ Γ( ρ ) , E ( u ) = inf { E ( v ) ; v ∈ Γ( ρ ) } has a solution, we shall denote by E ( ρ ) the set of all possible (constraint) energyminimizers. We call this set orbitally stable , if for all ε >
0, there exists δ > u ∈ H ( R ) satisfies inf φ ∈E ( ρ ) k u − φ k H δ, then the solution to (1.1) with u | t =0 = u satisfiessup t ∈ R inf φ ∈E ( ρ ) k u ( t, · ) − φ k H ε. Theorem 1.5 (Orbital stability of energy minimizers) . Given any ρ > , the set E ( ρ ) is non-empty and orbitally stable. R. CARLES AND C. SPARBER
The fact that energy minimizers are orbitally stable is in sharp contrast to thecase of the usual focusing cubic NLS in two spatial dimensions, for which all solitarywaves are known to be strongly unstable due to the possibility of blow-up, see [8].(In the defocusing case, there is no solitary wave and all solutions scatter.)Using re-arrangement inequalities, cf. [24], it is possible to infer that every energyminimizer is radially decreasing and solves (1.5) for some Lagrange multiplier ω > φ ω , possibly after anappropriate space translation (for a given mass and for a certain fixed ω , minimizingthe action or the energy is equivalent). The difficulty, however, is that several ω ’scould, at least in principle, yield the same mass ρ . Thus, uniqueness of solutionsto (1.5) at fixed ω does not imply the uniqueness of energy minimizers. The onlycases for which this uniqueness is known to be true seem to be the one of a singlepure power law nonlinearity | u | p − u , see [8], and the one of a purely logarithmicnonlinearity u ln | u | , cf. [1]. It is, nevertheless conjectured that uniqueness holdstrue for more general nonlinearities, see e.g. [7, 16, 23] for a more detailed discussionon this.The fact that there exists energy minimizer with arbitrarily small mass ρ > strictlybigger than the one of the cubic nonlinear ground state Q , see [7]. In Section 3.2,we shall present arguments showing that M ( φ ω ) ≡ k φ ω k L → , as ω → + .The rest of this paper is devoted to the proof of these theorems, which will bedone via a series of technical results given in Sections 2–4 below. In there, we willalso add further remarks and results on topics such as scattering and the asymptoticbehavior of φ ω . Finally, in an appendix, we address the analogue of (1.1) in 1D:Our Theorem A.1 suggests that ground states for (1.1) are indeed orbitally stablein the sense of, e.g., [12]. 2. Cauchy problem
Global well-posedness.
The aim of this subsection is to prove Theorem 1.1.To this end, we start by first proving local well-posedness of (1.1), when rewrittenthrough Duhamel’s formula, i.e.(2.1) u ( t ) = e i t ∆ u − iλ Z t e i t − s ∆ f ( u )( s ) ds, where here and in the following, we denote f ( z ) = z | z | ln | z | , z ∈ C . A classical fixed point argument, based on the use of Strichartz estimates, thenyields the following result.
Proposition 2.1 (Local well-posedness) . For any u ∈ H ( R ) and any λ ∈ R ,there exist times T > and a unique solution u ∈ C ([0 , T ]; H ( R )) ∩ C ((0 , T ); H − ( R )) , to (2.1) , depending continuously on u . Moreover u conserves its mass, energy,and momentum, and we also have the blow-up alternative, i.e. if T < ∞ , then lim t → T − k u ( t, · ) k H = ∞ . In view of the fact that (1.1) is time-reversible, we also obtain the analogousstatement backward in time.
NTERCRITICAL LOG-MODIFIED SCHR ¨ODINGER EQUATION 5
Proof.
We see that our nonlinearity f ∈ C ( R ; R ) satisfies f (0) = 0, | f ( u ) | . | u | − ε + | u | ε , ∀ ε > , as well as |∇ f ( u ) | (3 | ln | u | | + 2) | u | |∇ u | . ( | u | ε + | u | − ε ) |∇ u | . We therefore can simply quote classical results by Kato, in particular [19, The-orem I] (see also [8]), to obtain existence and uniqueness of a strong solution u ( t, · ) ∈ H ( R ) to (2.1), up to some (possibly finite) time T = T ( k u k H ) > u ). (cid:3) Remark . It is not clear whether the solution is arbitrarily smooth or not, ingeneral, since one can see that the third derivative of f ( z ) becomes singular. Seealso [6] in the case of the (even more singular) nonlinearity z ln | z | . Corollary 2.3 (Global well-posedness) . Let λ > . Then, the solution is global,i.e. T = ∞ .Proof. Using the conservation laws of mass and energy, together with the fact that λ >
0, the positive part of the energy satisfies E + ( u ) : = 12 k∇ u ( t, · ) k L + λ Z | u | > √ e | u ( t, x ) | ln (cid:18) | u ( t, x ) | √ e (cid:19) dx = E ( u ) + λ Z | u | < √ e | u ( t, x ) | ln (cid:18) √ e | u ( t, x ) | (cid:19) dx E ( u ) + λ Z R | u ( t, x ) | (cid:18) √ e | u ( t, x ) | (cid:19) dx = E ( u ) + λ √ eM ( u ) . This consequently yields a uniform in-time bound on k u ( t, · ) k H and thus, theblow-up alternative implies that T = ∞ . (cid:3) Some scattering results.
Let us introduce the conformal spaceΣ := (cid:8) f ∈ H ( R ) , x
7→ | x | f ( x ) ∈ L ( R ) (cid:9) , k f k Σ = k f k H ( R ) + k| x | f k L ( R ) . Lemma 2.4.
Let u ∈ Σ and λ > , then the global in-time solution u obtainedabove satisfies u ∈ C ( R ; Σ) .Proof. We introduce the Galilean operator J ( t ) = x + it ∇ , which commutes withthe free Schr¨odinger equation, i.e. (cid:2) J, i∂ t + ∆ (cid:3) = 0 . A direct computation then yields the pseudo-conformal conservation law ddt (cid:18) k ( x + it ∇ ) u k L + λt Z R | u ( t, x ) | ln (cid:18) | u ( t, x ) | √ e (cid:19) dx (cid:19) = − λt Z R | u ( t, x ) | dx. In particular if λ >
0, the same type of argument as in the proof above yieldsthat k ( x + it ∇ ) u k L is uniformly bounded for all t >
0. A triangle inequality thenimplies that u ( t, · ) ∈ Σ. (cid:3) Proposition 2.5.
Existence of wave operators: If u − ∈ Σ , then there exist u ∈ Σ and u ∈ C ( R ; Σ) solving (1.1) such that (cid:13)(cid:13)(cid:13) e − i t ∆ u ( t, · ) − u − (cid:13)(cid:13)(cid:13) Σ −→ t →−∞ . R. CARLES AND C. SPARBER
Small data scattering: If u ∈ Σ and k u k Σ is sufficiently small, then there exists u + ∈ Σ , such that (cid:13)(cid:13)(cid:13) e − i t ∆ u ( t, · ) − u + (cid:13)(cid:13)(cid:13) Σ −→ t →∞ . Sketch of the proof.
Recall that J ( t ) u = it e i | x | / (2 t ) ∇ (cid:16) ue − i | x | / (2 t ) (cid:17) , which implies that J ( t ) u can be estimated like ∇ u in L p . Using this, one obtainsthe Gagliardo–Nirenberg type inequality adapted to J ( t ), i.e. k u k L p ( R ) . t − /p k u k /pL ( R ) k ( x + it ∇ ) u k − /pL ( R ) , p < ∞ . Essentially the same fixed point argument as the one used in solving the Cauchyproblem locally in-time then yields the existence of wave operators (see e.g. [8]).Small data scattering then follows directly from [28, Theorem 2.1]. (cid:3)
Remark . The existence of wave operators under the mere assumption u − ∈ H ( R ) is very delicate, since the present nonlinearity can be understood as the sumof a slightly L -subcritical (focusing) nonlinearity and a slightly L -supercritical(defocusing) nonlinearity. The existence of wave operators in H is known for L -supercritical defocusing nonlinearities, but not for L -subcritical ones. Also, thesmallness in Σ is necessary to have scattering, in the sense that smallness in H ( R )is not enough, see also Remark 3.6.3. Nonlinear ground states
Necessary and sufficient conditions for the existence of ground states.
We seek solutions to (1.1) in the form u ( t, x ) = e iωt φ ( x ), with ω ∈ R and φ suffi-ciently smooth and localized. Then φ solves(3.1) − ∆ φ = g ( φ ) , on R , where here, and in the following, we shall denote (in agreement with the notationsfrom [2, 3]):(3.2) g ( φ ) = − ωφ − λ | φ | φ ln | φ | , G ( z ) := Z z g ( s ) ds. We also define the quantities T ( φ ) := Z R | φ ( x ) | dx, V ( φ ) := Z R G ( φ ( x )) dx, which allow us to rewrite the Lagrangian action as(3.3) S ( φ ) = 12 T ( φ ) − V ( φ ) . In a first step, we shall derive certain necessary conditions for solution φ to (3.1). Lemma 3.1 (Pohozaev identities) . Any solution φ ∈ H ( R ) to (3.1) satisfies (3.4) 12 Z R |∇ φ | dx + λ Z R | φ | ln | φ | dx + ω Z R | φ | dx = 0 , as well as (3.5) 12 Z R |∇ φ | dx + λ Z R | φ | dx = ω Z R | φ | dx. Moreover, in order to have a nontrivial solution φ , a necessary condition onthe frequency ω ∈ R is < ω < λ √ e . NTERCRITICAL LOG-MODIFIED SCHR ¨ODINGER EQUATION 7
Proof.
First, assume that φ is sufficiently smooth and rapidly decaying as | x | → ∞ .Then we directly obtain (3.4) by multiplying (3.1) with ¯ φ and integrating w.r.t. x ∈ R . To obtain (3.5), we instead multiply by (3.1) with x · ∇ ¯ φ . Integration in x then yields(3.6) λ Z R | φ | ln | φ | dx − λ Z R | φ | dx + ω Z R | φ | dx = 0 , or, in other words, V ( φ ) = 0. By taking (3.4) − × (3.6) we infer (3.5) for sufficiently“nice” φ , and a limiting argument allows us to extend this result to general φ ∈ H ( R ). In particular, (3.5) also implies that ω > φ .Next, we consider, for ε > c ε = sup
3) imposed in [2]. Tothis end, we first note that the function g ∈ C ( R ; R ) is obviously odd, and thatlim s → g ( s ) s = − ω < , since ω > . R. CARLES AND C. SPARBER
Thus ( g.
0) and ( g.
2) are indeed satisfied. In addition, we see that that g is sub-exponential at infinity, hence satisfying condition ( g. g. z > G ( z ) = − ωz − λ Z z s ln s ds = − ωz − λz ln z + λ z − ωz − λ z ln z √ e . The map z z − z ln z reaches its maximum at z ∗ = e − / , and G (cid:16) e − / (cid:17) = 1 √ e (cid:18) − ω + λ √ e (cid:19) > , by our assumption on ω . Therefore, also ( g.
2) is satisfied and we obtain our result.Finally, the exponential decay of the solution (together with its derivatives) followsfrom standard arguments for ordinary differential equations, see, e.g., [3, Section4.2]. (cid:3)
Uniqueness and further properties.
Having obtained existence of nonlin-ear ground states, we shall now derive further properties for them.
Lemma 3.3 ( L ∞ -bound) . Let φ ω be a nonlinear ground state. Then there existsa unique z ω ∈ ( e , , satisfying z ω → as ω → + , such that < φ ω ( x ) < √ z ω , for all x ∈ R ,Proof. In view of Proposition 3.2, we know that φ ω = φ ω ( r ) > φ ω (0)
0, thus λφ ω ln φ ω + ωφ ω | r =0 . Therefore, since φ ω (0) > z ln z − ωλ , where z = φ ω (0) . The map z z ln z is negative exactly on (0 , − e at z ∗ = e . Since ω ∈ (0 , λ √ e ) by assumption, there exists a unique z ω ∈ ( e , z ω ln z ω = − ωλ , and z ω → ω → (cid:3) Remark . The proof can be generalized to any sufficiently smooth solution φ to(3.1), not necessarily radial and decreasing. Indeed, the same argument as aboveshows that at any point x ∈ R where | φ | reaches its maximum: | φ ( x ) | √ z ω .Hence, the above estimate generalizes to | φ ( x ) | √ z ω , ∀ x ∈ R , as soon as φ ∈ C ( R ) solves (3.1). In particular, | φ ( x ) | < x ∈ R , henceln | φ | <
0, i.e., the nonlinearity can be considered fully focusing.We now turn to the question of uniqueness of nonlinear ground states.
Lemma 3.5 (Uniqueness) . There exists at most one positive solution φ ω to (3.1) .Proof. This result follows from [18, Theorem 1.1] provided we can check the condi-tion ( f − ( f
3) imposed on g . In view of (3.2), we see that g (0) = 0 and continuouson [0 , ∞ ). Recall that its anti-derivative is G ( z ) = λz (cid:18) z − z ln z − ωλ (cid:19) ≡ λz ˜ g ( z ) . NTERCRITICAL LOG-MODIFIED SCHR ¨ODINGER EQUATION 9
A straightforward calculation shows that ˜ g is strictly increasing on [0 , e − / ) andstrictly decreasing on ( e − / , ∞ ). In addition, we know that˜ g (0) = − ωλ < , ˜ g (cid:16) e − / (cid:17) > , and ˜ g ( z ) → −∞ , as z → + ∞ .Thus, we can choose u as the unique zero of ˜ g on the interval [0 , e − / ). Further-more we claim that we can choose ¯ u = √ z ω . To this end, one first checks that thereexists a unique α ∈ [0 , e − / ), such that g (0) = g ( α ) = g ( √ z ω ) = 0 and g ( z ) < , α ) ∪ ( √ z ω , ∞ ), while g ( z ) > α, √ z ω ).By the choice of u , we have that G ( u ) = Z u g ( z ) dz = 0 , and hence α < u . In particular, since g ( z ) > α, √ z ω ), this implies that G ( z ) > u , √ z ω ).Finally, to satisfy condition ( f s ( z ) = zg ′ ( z ) g ( z ) is decreasingon [ u , √ z ω ). This follows from a lengthy calculation which shows that g ( z ) s ′ ( z ) = 4 λz (cid:0) ω (1 + ln z ) − λz (cid:1) < , on ( u , √ z ω ). We therefore have all the necessary ingredients to conclude uniquenessof the ground state. (cid:3) The proof Theorem 1.3 is now complete.
Asymptotics for ω → . To show that M ( φ ω ) →
0, as ω →
0, one can followthe ideas in [21] for the cubic-quintic case (see also [27]). In there, the asymptoticregime ω → ψ ω ( x ) = 1 √ ω φ ω (cid:18) x √ ω (cid:19) , which is L ( R )-unitary. One finds that ψ ω solves − ∆ ψ ω + ωλψ ω − λψ ω + ψ ω = 0 , and thus, the limit ω → M ( ψ ω ) = M ( φ ω ) −→ ω → M ( Q ) , where Q is the cubic ground state solution to −
12 ∆ Q − λQ + Q = 0 , In our case, the logarithm is not compatible with such a rescaling. Instead, wedefine ψ ω ( x ) = s ln ω ω φ ω (cid:18) x √ ω (cid:19) , and a computation shows that ψ ω solves −
12 ∆ ψ ω − λψ ω + ψ ω = λ ln ln ω ln ω ψ ω − λ ln ω ψ ω ln ψ ω . Recalling that, as ω → ≪ ln ln 1 ω ≪ ln 1 ω , and using the analyticity of ψ ω in ω , we have ψ ω ∼ ω → Q , and thus, in terms of φ ω , φ ω ( x ) ∼ ω → r ω ln ω Q ( x √ ω ) . In turn, this implies that M ( φ ω ) = 1 q ln ω M ( Q ) −→ ω → . These formal arguments can be made rigorous by following the steps in [21], whichare based on the use of the linearized operator L : f
7→ − f − λQ f + f, which is known to be an isomorphism L : H → H − , cf. [34]. The implicitfunction theorem then allows one to write ψ ω in terms of Q plus lower order cor-rections involving L − . In the present case, the situation is similar, for the spectralanalysis presented in [21] is readily adapted to the present case. Details are left tothe interested reader. Remark . This computation also shows that the L ∞ -bound derived before is farfrom being sharp for small ω . The fact that the L -norm of φ ω can be arbitrarilysmall, is in sharp contrast with the cubic-quintic case. Also, (3.5) shows thatthe H -norm of φ ω can be arbitrarily small: smallness in H does not guaranteescattering. The smallness of the momentum in [28, Theorem 2.1] (and thus k u k Σ sufficiently small) must be considered as necessary (since φ ω decays exponentially).4. Orbital Stability
We start by recalling that for ρ > ρ ) = (cid:8) u ∈ H ( R d ) , M ( u ) = ρ (cid:9) , and first prove that the constrained energy is bounded below. Lemma 4.1 (Bound on the energy) . For any ρ > , inf { E ( u ) ; u ∈ Γ( ρ ) } = − ν, for some finite ν > .Proof. We can estimate E ( u ) > k∇ u k L − λ Z | u | < √ e | u | ln (cid:18) √ e | u | (cid:19) dx > k∇ u k L − λ √ e Z R | u | dx > k u k H − K, where K = ρ (1 + λ √ e ) >
0. Thus, all (constrained) energy-minimizing sequencesare bounded in H ( R ) and − ν > − K > −∞ . Moreover, for µ >
0, let u µ ( x ) := µu ( µx ) such that k u µ k L ( R ) = k u k L ( R ) .Then, one finds that E ( u µ ) = µ E ( u ) − µ λ ln (cid:18) µ (cid:19) Z R | u | dx. Hence, E ( u µ ) < µ > ν > (cid:3) We shall now show that energy minimizers indeed exist, and that they are or-bitally stable (as a set), by invoking the concentration-compactness method of [25](see also [8, Proposition 1.7.6].
Proof of Theorem 1.5.
We proceed in several steps:
NTERCRITICAL LOG-MODIFIED SCHR ¨ODINGER EQUATION 11
Step 1.
Let ( u n ) n > ⊂ H ( R ) be a minimizing sequence to (1.6). In view of[25], we have the standard trichotomy of concentration compactness. To rule outvanishing of the sequence, we first note that for n sufficiently large, Lemma 4.1implies that E ( u n ) − ν , and hence, from the proof of Lemma 4.1, Z | u n | < √ e | u n | ln (cid:18) √ e | u n | (cid:19) dx > νλ > . In addition, Z | u n | < √ e | u n | dx & Z | u n | < √ e | u n | ln (cid:18) √ e | u n | (cid:19) dx, and, thus, any minimizing sequence is bounded away from zero in L ( R ). Step 2.
Next, we need to rule out dichotomy, in order to conclude compactness.Arguing by contradiction, suppose that, after the extraction of some suitable sub-sequences, there exist ( v k ) k > , ( w k ) k > in H ( R ), such thatsupp v k ∩ supp w k = ∅ , as well as the following properties: k v k k L −→ k →∞ θρ, k w k k L −→ k →∞ (1 − θ ) ρ, for some θ ∈ (0 , , (4.1) lim inf k →∞ (cid:18)Z |∇ u n k | − Z |∇ v k | − Z |∇ w k | (cid:19) > , and the remainder r k := u n k − v k − w k satisfies k r k k L p −→ k →∞ , for all 2 p < ∞ . Note that this also implies (cid:12)(cid:12)(cid:12)(cid:12)Z | u n k | p − Z | v k | p − Z | w k | p (cid:12)(cid:12)(cid:12)(cid:12) −→ k →∞ , since v k and w k have disjoint support.Denote h ( y ) = y ln y for y >
0. A Taylor expansion on h ( y + z ) − h ( y ) − h ( z ),combined with an induction step shows that for ε > N >
1, that there existsa C ε,N >
0, such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h N X n =1 y n ! − N X n =1 h ( y n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ε,N N X ℓ = k | y ℓ | (cid:0) | y k | − ε + | y k | ε (cid:1) . Applying this with ε = 1 and N = 3 to v k , w k , r k , and integrating over R , yields (cid:12)(cid:12)(cid:12)(cid:12)Z h ( u n k ) − Z h ( v k ) − Z h ( w k ) (cid:12)(cid:12)(cid:12)(cid:12) . Z h ( r k ) ++ Z | r k | (cid:0) | v k | + | v k | + | w k | + | w k | (cid:1) + Z ( | v k | + | w k | ) (cid:0) | r k | + | r k | (cid:1) , where in the second line we have used the fact that v k and w k have disjoint supports.Applying H¨older’s inequality and recalling that k r k k L p →
0, as k → ∞ , shows thatall the integrals on the right hand side tend to zero in the limit k → ∞ , hence (cid:12)(cid:12)(cid:12)(cid:12)Z h ( u n k ) − Z h ( v k ) − Z h ( w k ) (cid:12)(cid:12)(cid:12)(cid:12) −→ k →∞ . Recalling that Z | u n k | − Z | v k | − Z | w k | −→ k →∞ , we obtain Z | u n k | ln (cid:18) | u n k | √ e (cid:19) − Z | v k | ln (cid:18) | v k | √ e (cid:19) − Z | w k | ln (cid:18) | w k | √ e (cid:19) −→ k →∞ . We consequently infer from (4.1) thatlim inf k →∞ ( E ( u n k ) − E ( v k ) − E ( w k )) > , and thus(4.2) lim sup k →∞ ( E ( v k ) + E ( w k )) − ν. Following an idea from [11], we now use a scaling argument and set˜ v k ( x ) = v k (cid:16) σ − / k x (cid:17) , σ k = ρ k v k k L ˜ w k ( x ) = w k (cid:16) µ − / k x (cid:17) , µ k = ρ k w k k L . We have M (˜ v k ) = M ( ˜ w k ) = ρ , and hence E (˜ v k ) , E ( ˜ w k ) > − ν. We also find that E (˜ v k ) = σ k (cid:18) σ k Z |∇ v k | − λ Z | v k | ln (cid:18) | v k | √ e (cid:19)(cid:19) , and so E ( v k ) = 1 σ k E (˜ v k ) + 1 − σ − k Z |∇ v k | > − νσ k + 1 − σ − k Z |∇ v k | . Doing the same for E ( w k ), yields E ( v k ) + E ( w k ) > − ν (cid:18) σ k + 1 µ k (cid:19) + 1 − σ − k Z |∇ v k | + 1 − µ − k Z |∇ w k | > − ν (cid:18) σ k + 1 µ k (cid:19) + 1 − σ − k k v k k L k v k k L + 1 − µ − k k w k k L k w k k L , where in the second step, we have used the Gagliardo-Nirenberg inequality. Passingto the limit, we inferlim inf k →∞ ( E ( v k ) + E ( w k )) > − ν + 12 min (cid:18) − θθρ , θ (1 − θ ) ρ (cid:19) lim inf k →∞ k u n k k L , for any θ ∈ (0 , k u k L k u k L k u k L and thus, in view ofStep 1 and the fact that k u n k k L = ρ >
0, we inferlim inf k →∞ k u n k k L > . This is in contradiction to (4.2) and consequently rules out dichotomy.
Step 3.
We can now invoke [8, Proposition 1.7.6(i)] to deduce that for u ∈ H ( R )and ( y k ) ⊂ R : u n k ( · − y k ) → u in L p ( R ) for all 2 p < ∞ . Together with theweak lower semicontinuity of the H norm and the usual bound on the nonlinearpotential energy, this implies E ( u ) lim k →∞ E ( u n k ) = − ν, and thus, the existence of a constraint energy minimizer. Step 4.
The orbital stability now follows by invoking classical arguments of [9](see also [8]): Assume, by contradiction, that there exist a sequence of initial data( u ,n ) n ∈ N ⊂ H ( R ), such that(4.3) k u ,n − φ k H −→ n →∞ , NTERCRITICAL LOG-MODIFIED SCHR ¨ODINGER EQUATION 13 and a sequence ( t n ) n ∈ N ⊂ R , such that the sequence of solutions u n to (1.1) asso-ciated to the initial data u ,n satisfies(4.4) inf ϕ ∈E ( ρ ) k u n ( t n , · ) − ϕ k H > ε, for some ε >
0. Denoting v n = u n ( t n , · ), the above inequality readsinf ϕ ∈E ( ρ ) k v n − ϕ k H > ε. In view of (4.3), we find that, one the one hand: Z R | u ,n | −→ n →∞ Z R | φ | , E ( u ,n ) −→ n →∞ E ( φ ) = inf v ∈ Γ( ρ ) E ( v ) . One the other hand, the conservation laws for mass and energy imply M ( v n ) −→ n →∞ M ( φ ) , E ( v n ) −→ n →∞ E ( φ ) , and thus, ( v n ) n is a minimizing sequence for the problem (1.6). From the previoussteps, there exist a subsequence, still denoted by ( u n ) n ∈ N , and a sequence of points( y n ) n ∈ N ⊂ R , such that v n ( · − y n ) has a strong limit u in H ( R ). In particular, u satisfies (1.6), hence a contradiction. (cid:3) Appendix A. On the 1D case
Since the L -critical case in 1D requires a quintic nonlinearity, the formal ana-logue of (1.1) reads(A.1) i∂ t u + 12 ∂ x u = λu | u | ln | u | , x ∈ R , λ > ,u (0 , x ) = u ∈ H ( R ) . Even though, to our knowledge, this model is not motivated by physics, it is math-ematically similar and gives a hint of what could be expected for (1.1). Globalwell-posedness follows from the same arguments as in Theorem 1.1. The analogueof Theorem 1.3 is also straightforward, and yields the condition 0 < ω < λ e / ,since, in view of [3, Theorem 5], we compute G ( z ) = − ωz − λ z ln z e / . We then have a stronger notion of orbital stability than in the case of Theorem 1.5:
Theorem A.1.
Let < ω < λ e / , and φ ω be the unique even and positive solutionto − φ ′′ ω + λφ ω | φ ω | ln | φ ω | + ωφ ω = 0 , x ∈ R . Then, for all ε > , there exists δ > such that if u ∈ H ( R ) satisfies k u − φ k H ( R ) δ, the solution to (A.1) satisfies sup t ∈ R inf θ ∈ R y ∈ R (cid:13)(cid:13) u ( t, · ) − e iθ φ ω ( · − y ) (cid:13)(cid:13) H ( R ) ε. Proof.
The proof relies on the Grillakis-Shatah-Strauss theory [15], following thebreakthrough of M. Weinstein [35] (see also [12]), which implies that result is provenif we know that d ( ω ) := S ( φ ω ) is strictly convex, or, equivalently, that M ( φ ω ) isstrictly increasing. Taking advantage of the one-dimensional setting, Iliev andKirchev [17, Lemma 6] have shown that d ′′ ( ω ) = − W ′ ( a ) Z a (cid:18) as ( f ( a ) − f ( s )) ag ( s ) − sg ( a ) (cid:19) (cid:18) sW ( s ) (cid:19) / ds, where, in the present case, f ( s ) = λs ln s, g ( s ) = Z s f ( σ ) dσ = λ s ln (cid:16) se / (cid:17) , W ( s ) = ωs + g ( s ) , and a is such that W ( a ) = 0, W ′ ( a ) <
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Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
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Department of Mathematics, Statistics, and Computer Science, M/C 249, Universityof Illinois at Chicago, 851 S. Morgan Street Chicago, IL 60607, USA
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