Nonlinear Schr{ö}dinger equation: concentration on circles driven by an external magnetic field
aa r X i v : . [ m a t h . A P ] S e p NONLINEAR SCHRÖDINGER EQUATION: CONCENTRATION ON CIRCLESDRIVEN BY AN EXTERNAL MAGNETIC FIELD
DENIS BONHEURE, SILVIA CINGOLANI, AND MANON NYS
Abstract.
In this paper, we study the semiclassical limit for the stationary magnetic nonlinearSchrödinger equation ( i ~ ∇ + A ( x )) u + V ( x ) u = | u | p − u, x ∈ R , (0.1)where p > A is a vector potential associated to a given magnetic field B , i.e ∇ × A = B and V isa nonnegative, scalar (electric) potential which can be singular at the origin and vanish at infinityor outside a compact set. We assume that A and V satisfy a cylindrical symmetry. By a refinedpenalization argument, we prove the existence of semiclassical cylindrically symmetric solutions of(0 .
1) whose moduli concentrate, as ~ →
0, around a circle. We emphasize that the concentration isdriven by the magnetic and the electric potentials. Our result thus shows that in the semiclassicallimit, the magnetic field also influences the location of the solutions of (0 .
1) if their concentrationoccurs around a locus, not a single point.2010
AMS Subject Classification.Keywords
Nonlinear Schrödinger equation; semiclassical states; Singular potential; Vanishing poten-tial; Concentration on curves; External magnetic field; Variational methods; penalization method. Introduction
In Quantum Mechanics, the nonlinear Schrödinger equation (NLS) with a exterior magnetic field B , having source in the magnetic potential A , and a scalar (electric) potential U has the form i ~ ∂ψ∂t = ( i ~ ∇ + A ( x )) ψ + U ( x ) ψ = f ( | ψ | ) ψ, x ∈ R N , where N ≥ i = − ~ is the Planck constant, and the mass is taken m = 1 / i ~ ∇ + A ( x )) = − ~ ∆ + 2 i ~ A ( x ) · ∇ + i ~ div A ( x ) + | A ( x ) | , and f ( | ψ | ) ψ is a nonlinear term. In dimension N = 3, the magnetic potential A is related to themagnetic field B by the relation B = ∇ × A . Such evolution equation arises in various physicalcontexts, such as nonlinear optics or plasma physics, where one simulates the interaction effectamong many particles by introducing a nonlinear term.The search of standing waves ψ ( x, t ) = e − i E ~ t u ( x ) leads to study the stationary nonlinear mag-netic Schrödinger equation( i ~ ∇ + A ( x )) u + ( U ( x ) − E ) u = f ( | u | ) u, x ∈ R N . (1.1) Date : September 7, 2018.D.B. is supported by INRIA - Team MEPHYSTO, MIS F.4508.14 (FNRS), PDR T.1110.14F (FNRS) & ARCAUWB-2012-12/17-ULB1- IAPAS. He thanks the support of INDAM during his visits at the Politecnico di Bari wherepart of this work has been done. S.C. is partially supported by GNAMPA-INDAM Project 2015
Analisi variazionaledi modelli fisici non lineari.
She thanks the support of FNRS during her visit at Université libre de Bruxelles wherepart of this work has been done. M.N. is a Research Fellow of the Belgian Fonds de la Recherche Scientifique - FNRSand is partially supported by the project ERC Advanced Grant 2013 n. 339958: “Complex Patterns for StronglyInteracting Dynamical Systems -COMPAT”. She thanks the Università di Torino for hospitality.
In the following, we write V ( x ) = U ( x ) − E and for simplicity we consider f ( t ) = t ( p − / . However,we note that a larger class of nonlinearity could be considered, see for instance [34].For ~ > | u ~ | vanishes at infinity was firstproved by Esteban and Lions in [22] by using a constrained minimization approach. Concentrationand compactness arguments are applied to solve the associated minimization problems for a broadclass of magnetic fields. Successively in [5], Arioli and Szulkin studied the existence of infinitelymany solutions of (1.1) assuming that V and B are periodic.In the present paper, we are interested in the semiclassical analysis of the magnetic nonlinearSchrödinger equation (1.1). From a mathematical point of view, the transition from quantum toclassical mechanics can be formally performed by letting ~ →
0. For small values of ~ >
0, solutions u ~ : R N → C of (1.1) are usually referred to as semiclassical (ground or bound) states.When A = 0, the study of the nonlinear Schrödinger equation − ~ ∆ u + V ( x ) u = f ( | u | ) u, x ∈ R N , (1.2)has been extensively pursued in the semiclassical regime and a considerable amount of work has beendone, showing that existence and concentration phenomena of single- and multiple-spike solutionsoccur at critical points of the electric potential V when ~ →
0, see e.g. [1, 3, 14, 15, 19, 20,23, 32, 34]. Successively, the question of existence of semiclassical solutions to NLS equationsconcentrating on higher dimensional sets has been investigated. In [4] Ambrosetti, Malchiodi andNi considered the case of a radial potential V ( | x | ) and constructed radial solutions exhibitingconcentration on a sphere, which radius is a non degenerate critical point of the concentrationfunction M ( r ) = r N − V σ ( r ), σ = p/ ( p − − / ~ n →
0, whenever the sphere is replacedby a closed hypersurface Γ, stationary and non degenerate for the weighted area functional R Γ V σ .In [21], the above conjecture was completely solved in the plane by Del Pino, Kowalczyk and Wei.We also quote [28], where Malchiodi and Montenegro considered the NLS equation on a smoothbounded domain Ω in R with Neumann boundary conditions and proved, for a suitable sequence ~ n →
0, the existence of positive solutions u ~ n concentrating at the whole boundary of Ω or atsome components of it. In [26, 27], boundary concentration on a geodesic of the boundary has beentreated in the three-dimensional case. Later on, concentration on spheres of dimension N − k -dimensional spheres ( k ≤ ≤ N − k -dimensional sphere, 1 ≤ k ≤ N −
1, for a large class of symmetric potentials V .In presence of a magnetic field ( A = 0), a challenging question is to establish how the magneticfield influences the existence and the concentration of the moduli of the complex-valued solutions of(1.1) as ~ →
0. A first result dealing with the concentration of least-energy solutions for magneticNLS equations was obtained in [25]. In this paper, Kurata proved that if ( u ~ ) ~ is a sequence ofleast-energy solutions to (1.1) with f ( t ) = t ( p − / , then the sequence ( | u ~ | ) ~ of their moduli mustconcentrate at a global minimum x of V , as ~ →
0. More precisely, there exist a sequence of points( x n ) n ⊂ R N and a subsequence still denoted by ( ~ n ) n , with x n → x and ~ n → n → + ∞ , suchthat v ~ n ( y ) = u ~ n ( x n + ~ n y ) converges to some v ∈ C and converges also weakly in L p . Moreover, v satisfies the limiting equation( i ∇ + A ( x )) v + V ( x ) v = | v | p − v, x ∈ R N . If we let w ( x ) = e − iA ( x ) · x v ( x ), it follows that w satisfies weakly the equation − ∆ w + V ( x ) w = | w | p − w, x ∈ R N . ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 3
Hence the concentration of the least-energy solutions is driven by the electric potential while themagnetic potential influences the phase factor of the solutions, but does not affect the location ofthe peaks of their moduli. The existence of such semiclassical least-energy solutions for magneticNLS equations was established in [11] by using Ljusternick-Schnirelmann theory.Successively, by using a penalization argument, the existence of semiclassical bound state so-lutions to (1.1), concentrating at local minima of V , has been proved in [17], for a large class ofmagnetic potentials, covering the case of polynomial growths corresponding to constant magneticfield (see also [16] for bounded potentials).We also refer to [13] for existence results of multi-peak solutions to (1.1), whose moduli havemultiple concentration points around local minima of V , dealing with a large class of nonlinearterms (possibly not monotone); and to [12] for semiclassical solutions having specific symmetriesconcentrating around orbits of critical points of V .In [35], the authors have established necessary conditions for a sequence of standing wave so-lutions of (1.1) to concentrate, in different senses, around a given point. More precisely, theyshow that if f ( t ) = t ( p − / , then the moduli of the peaks have to locate at critical points of V ,independently of A , confirming what was conjectured in [16].In all the above cited papers, the concentration of the moduli of the complex valued solutionsoccurs at one or a finite set of critical points of the electric field V , while the magnetic field onlyinfluences the phase factor of the standing waves as ~ is small.In the present paper, we are interested in studying concentration phenomena on higher dimen-sional sets in the presence of a magnetic field. More particularly, we aim to understand how andin which situations the magnetic field influences such concentration. In the following, we restrictourself to consider (1.1) in R , for which we can already detect some interesting phenomena.More specifically, we consider the class of scalar potentials V invariant under a group G oforthogonal transformations, and the class of magnetic potentials A equivariant under the samegroup, that is g A ( g − x ) = A ( x ) , (1.3)for every g ∈ G .In dimension 3, the simplest group is G = O (3) which corresponds to a radially symmetricsetting. The potential V then depends only on | x | , while A satisfies the equivariance condition(1.3) for every g ∈ O (3). However, this last constraint on A is too strong, in the sense that the onlypossible vector potential satisfying this condition is a multiple of the normal vector to the sphere.Indeed, if x is a point on a sphere of radius r , there always exist rotations g x ∈ O (3) that leave theaxis going through the center of the sphere and x invariant, that is g x x = x for those particular g x .Then, at that point x , the equivariance condition (1.3) rewrites g x A ( g − x x ) = A ( x ) ⇒ g x A ( x ) = A ( x ) . This means that at that point x , A ( x ) = f ( x ) x , where f ( x ) is any arbitrary function of x . Finally,if we consider any g ∈ O (3) with A having the above expression, we obtain g f ( g − x ) (cid:16) g − x (cid:17) = f ( g − x ) x = f ( x ) x, for all g ∈ O (3) . This means that A ( x ) = f ( r ) x is a normal vector to the sphere, depending only of the radius of thesphere. Furthermore, we immediately notice that A is a conservative field and therefore ∇ × A = B = 0. We remark that this result was already obtained in [18, Theorem 1.3]. Then physically,(1.1) is equivalent to a problem without magnetic potential. In particular, the concentration onspheres of the solutions of (1.1) is only driven by the scalar potential V and we are exactly on thecase studied by Ambrosetti, Malchiodi and Ni in [4]. DENIS BONHEURE, SILVIA CINGOLANI, AND MANON NYS
A physically relevant case occurs in R in presence of magnetic and electric potentials havingcylindrical symmetries. In that setting, we obtain a new surprising result for (1.1). We prove thatthe existence and the concentration of semiclassical bound states is influenced by the magneticfield when the concentration occurs on a circle. We conjecture that this result should also occur inmore general situations. More specifically, we consider the class of invariant scalar potentials andequivariant magnetic potentials under the action of the group G := { g α ∈ O (3) , α ∈ [0 , π [ } , (1.4)where g α = cos α − sin α α cos α
00 0 ± . Namely, we assume that A = ( A , A , A ) ∈ C ( R , R ) satisfies (1.3) for every g ∈ G given by(1.4). If we use the cylindrical coordinates ( x , x , x ) = ( ρ cos θ, ρ sin θ, x ), the condition (1.3) canbe rewritten as A ( ρ, θ − α, ± x ) = cos α A ( ρ, θ, x ) + sin α A ( ρ, θ, x ) A ( ρ, θ − α, ± x ) = − sin α A ( ρ, θ, x ) + cos α A ( ρ, θ, x ) A ( ρ, θ − α, ± x ) = ± A ( ρ, θ, x ) . If we denote by e τ = ( − sin θ, cos θ, , e n = (cos θ, sin θ, , e = (0 , , R , we therefore infer that A has the form A ( ρ, θ, x ) = φ ( ρ, | x | ) e n + c ( ρ, | x | ) e τ + A ( ρ, x ) e , for some functions φ, c ∈ C ( R + × R + ) and some A ∈ C ( R + × R ) which is odd in x . The typicalexample φ ≡ ≡ A and c = bρ/ b ∈ R \{ } corresponds to the constant magnetic field B = b inthe direction x which is the simplest but also one of the more relevant case.Next, we consider nonnegative cylindrically invariant potentials V ∈ C ( R \{ } ), i.e. V ( gx ) = V ( x ) for every g ∈ G . This is equivalent to assume that V depends only on ρ and | x | . Moreover,we impose a growth condition at infinity when p ∈ (2 ,
4] :( V ∞ ) there exists α ≤ | x |→ + ∞ V ( x ) | x | α > p >
4, we do not impose this restriction so that for instance one can deal with fast-decayingpotentials or even compactly supported potentials. Nonetheless, as we will see later, we cannotconsider V ≡ V . For instance, V can behavesingularly at the origin, or be locally bounded at the origin. However, if V has a singularity, onecan single out the Hardy potential as a threshold behaviour as in [7]. If we assume in addition that( V ) there exists α ≥ | x |→ V ( x ) | x | α > u : R → C of the problem ( ( i ~ ∇ + A ) u + V u = | u | p − u,u ∈ L ( R , C ) , ( i ~ ∇ + A ) u ∈ L ( R , C ) . (1.5) ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 5 which satisfy the condition u ( gx ) = u ( x ) for all g ∈ G, x ∈ R , and concentrate around a circle for small ~ >
0. To this aim, we introduce the concentrationfunction M : R + × R + → R + defined by M ( ρ, | x | ) = 2 πρ h c ( ρ, | x | ) + V ( ρ, | x | ) i p − E (0 , , (1.6)where E (0 ,
1) is a positive unrelevant constant (see Section 3 for more details).Denoting by
H ⊂ R the 1-dimensional vectorial subspace spanned by e , and by H ⊥ its orthog-onal complement, we assume the existence of a smooth bounded open G -invariant set Λ ⊂ R suchthat ¯Λ ∩ H = ∅ , Λ ∩ H ⊥ = ∅ . By G -invariant, we mean that one has g (Λ) = Λ for every g ∈ G .Furthermore we assume thatinf Λ ∩H ⊥ M < inf ∂ Λ ∩H ⊥ M and inf Λ ∩H ⊥ M < Λ M , (1.7)whereas inf ¯Λ V > . (1.8)Observe that the second assumption in (1.7) is in fact not restrictive if we take Λ sufficiently small,since Λ is smooth and V is continuous in ¯Λ.Using the facts that A and V have cylindrical symmetries, the equation in (1.5) can be reducedto a problem in R . Let ρ > A : R → R the constant magneticpotential defined by A = ( φ ( ρ , ,
0) and a = c ( ρ ,
0) + V ( ρ , . We introduce the following two-dimensional problem( i ∇ + A ) u + a u = | u | p − u, y = ( y , y ) ∈ R (1.9)which can be regarded as a limiting problem for (1.5). Following the approach in [7], we will obtainthe existence of cylindrically symmetric solutions of (1.1) concentrating around circles in Λ ∩ H ⊥ for ~ > M , takes into account the magnetic field which will therefore influence the location of the concen-tration set of the semiclassical solutions of (1.1). This feature is new and, up to our knowledge,different from all the previous results in literature when dealing with an exterior magnetic field.Moreover, if we let A τ ( ρ ) = A ( ρ, θ, · e τ = c ( ρ,
0) be the tangential component of A ( ρ, θ,
0) and A n ( ρ ) = A ( ρ, θ, · e n = φ ( ρ,
0) be its normal component, the solution of the two-dimensional limitproblem (1.9) is given by e i ( A n ( ρ ) , · y w , y ∈ R , where w is the ground state solution of − ∆ w + a w = | w | p − w, y ∈ R . In this equation, a = c ( ρ ,
0) + V ( ρ ,
0) = A τ ( ρ ) + V ( ρ , A and by the scalar potential V , while the phase factor of the semiclassicalwave depends on the normal component of A . We conjecture that this is a general fact and that itis not just a consequence of the symmetry assumptions.In order to state our main result, we introduce some notations and tools adapted to the cylindricalsymmetry of the problem. First, for y, z ∈ R , we define the pseudometricd cyl ( y, z ) = (cid:16) ( ρ y − ρ z ) + ( y − z ) (cid:17) / , DENIS BONHEURE, SILVIA CINGOLANI, AND MANON NYS where ρ y = ( y + y ) / and ρ z = ( z + z ) / . This function accounts for the distance between twocircles. Then, for r > x ∈ R , we denote by B cyl ( x, r ) the ball (which is torus shaped) B cyl ( x, r ) = n y ∈ R | d cyl ( x, y ) < r o . Our main theorem states, for ~ sufficiently small, the existence of solutions of (1 .
5) that con-centrate around a circle S ~ in the plane x = 0, centered at the origin and of radius ρ ~ , where ρ ~ converges to a minimizer of M in Λ ∩ H ⊥ . Theorem 1.1.
Let p > . Let V ∈ C ( R \{ } ) and A ∈ C ( R , R ) be such that V ( gx ) = V ( x ) and g A ( g − x ) = A ( x ) , for every g ∈ G defined in (1.4) . Moreover, if p ∈ (2 , , we suppose V satisfies ( V ∞ ) . Assume that there exists a bounded smooth G -invariant set Λ ⊂ R such that (1.7) and (1.8) are satisfied. Then there exists ~ > such that for every < ~ < ~ (i) the problem (1 . has at least one solution u ~ ∈ C ,α loc ( R \{ } ) such that u ~ ( gx ) = u ~ ( x ) forevery g ∈ G .Moreover, for every < ~ < ~ , | u ~ | attains its maximum at some x ~ = ( ρ ~ cos θ, ρ ~ sin θ, x , ~ ) ⊂ Λ , θ ∈ [0 , π [ , such that (ii) lim inf ~ → | u ~ ( x ~ ) | > . (iii) lim ~ → M ( x ~ ) = inf Λ ∩H ⊥ M ; (iv) lim sup ~ → d cyl ( x ~ , H ⊥ ) ~ < + ∞ , that is x , ~ → ; (v) lim inf ~ → d cyl ( x ~ , ∂ Λ) > .Finally, for every < ~ < ~ there exist C > and λ > such that the following asymptotic holds (vi) 0 < | u ~ ( x ) | ≤ C exp − λ ~ d cyl ( x, x ~ )1 + d cyl ( x, x ~ ) ! (1 + | x | ) − ∀ x ∈ R \ { } . Remark 1.2.
The last assertion (vi) in Theorem 1.1 combines a concentration estimate with adecay as | x | → ∞ . This decay at infinity is not enough to guarantee that our solutions are L (sincethe ambiant space is R ). However, this is only a rough estimate valid without further assumptionon V and it can be improved when assuming a slow decay of V at infinity. Namely if we assumethat ( V ∞ ) holds, then the solutions decay fast enough to be square integrable and thus they are truebounded state solutions. We mention also that when ( V ) holds, we can estimate the flatness of thesolution at the origin. We refer to Lemma 6.5 for more details. Example 1.3.
As a striking example, we observe that the presence of a constant magnetic field canproduce a concentration phenomenon when coupled with a decaying electric potential. If we considerfor instance the cubic nonlinearity, i.e. p = 4 , and the cylindrical Hardy potential V ( ρ ) = 1 /ρ , ρ = x + x , there is no concentrated bound state (probably no bound state at all) of the equationwithout magnetic field. The presence of a constant magnetic field B = b in the direction x producesa solution that concentrates on the circle of radius / / (3 b ) / . As a particular case of Theorem 1.1, we deduce the somewhat surprising result which states thatwhen the scalar potential V is constant, the existence and the location of our semiclassical statesis only driven by the magnetic field. Corollary 1.4.
Assume V ≡ ω , where ω is a positive constant. Then, under the assumptions ofTheorem 1.1, the concentration of the solutions u ~ holds at inf Λ ∩H ⊥ M = inf Λ ∩H ⊥ πρ (cid:16) c + ω (cid:17) p − E (0 , . ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 7
Finally, we also remark that Theorem 1.1 do not require an upper bound on p >
2. Henceforth,we can treat critical and supercritical exponent problems by looking for cylindrically symmetricsolutions of (1.1).The paper is organized as follows. In Section 2, we give the variational framework and somerelated properties. Section 3 is devoted to the study of the two dimensional limit problem. Apenalization scheme is introduced in Section 4 and the existence of least-energy solutions is proved.The asymptotics of those solutions is studied in Section 5, while their concentration behaviour isestablished in Section 6, showing that the solutions of the penalized problem solve the original oneand therefore concluding the proof of Theorem 1.1. Finally, Section 7 is devoted to another classof symmetric solutions in the special case of a Lorentz type magnetic potential. These solutionsare defined through an ansatz proposed by Esteban and Lions in [22, Section 4.3].2.
The variational framework
In this section, we will fix our functional setting. In particular, we will define the Hilbert spacesadapted to the presence of a magnetic potential. We emphasize that in all Hilbert spaces we use,the scalar product will always be taken as the real scalar product, i.e. for every z, w ∈ C , the scalarproduct will be defined by ( w | z ) = Re( w ¯ z ).2.1. The magnetic spaces.
Let N ≥
2. For A ∈ L ( R N , R N ), we define the space D , A,ε ( R N , C )as the closure of C ∞ ( R N , C ) with respect to the norm defined through k u k D , A,ε := Z R N | ( iε ∇ + A ) u | . Similarly, D , ( R N , C ) (resp. D , ( R N , R )) is the closure of C ∞ ( R N , C ) (resp. C ∞ ( R N , R )) withrespect to the norm defined through k u k D , := Z R N |∇ u | . Remember that the Sobolev inequality implies that D , ( R N , C ) (resp. D , ( R N , R )) is embeddedin L ⋆ ( R N , C ) (resp. L ⋆ ( R N , R )). We also consider the space H A,ε ( R N , C ) = n u ∈ L ( R N , C ) | ( iε ∇ + A ) u ∈ L ( R N , C N ) o , endowed with the norm k u k H A,ε = Z R N | ( iε ∇ + A ) u | + | u | . We remark that, in general, this space is not embedded in H ( R N , C ) (and inversely). However if u ∈ H A,ε ( R N , C ), then | u | ∈ H ( R N , R ). This is the diamagnetic inequality that we recall here. Lemma 2.1 (Diamagnetic inequality) . Let A : R N → R N be in L loc ( R N , R N ) and let u ∈D , A,ε ( R N , C ) . Then, | u | ∈ D , ( R N , R ) and the diamagnetic inequality ε |∇| u | ( x ) | ≤ | ( iε ∇ + A ) u ( x ) | (2.1) holds for almost every x ∈ R N and for every ε > .Proof. We compute ε ∇| u | = Im (cid:18) iε ∇ u ¯ u | u | (cid:19) = Im (cid:18) ( iε ∇ + A ) u ¯ u | u | (cid:19) a.e.because A is real-valued. We conclude using the fact that | Im( z ) | ≤ | z | for any complex number z . (cid:3) DENIS BONHEURE, SILVIA CINGOLANI, AND MANON NYS
Using (2.1), we can verify that, for every u ∈ D , A,ε ( R N , C ), ε Z R N |∇| u || d x ≤ Z R N | ( iε ∇ + A ) u | d x, (2.2)for any ε > H A,ε ( R N , C ) and H ( R N , C ) are equivalent as for instanceif A is bounded. The following lemma is proved for example in [16, Lemma 3.1]. Lemma 2.2.
Let A : R N → R N be such that | A | ≤ C for all x ∈ R N , C ≥ . Then, the spaces H A,ε ( R N , C ) and H ( R N , C ) are equivalent. Finally, we introduce the following Hilbert space H A,V,ε ( R N , C ) = (cid:26) u ∈ D , A,ε ( R N , C ) | Z R N V ( x ) | u | < ∞ (cid:27) , endowed with the norm k u k H A,V,ε = Z R N | ( iε ∇ + A ) u | + V ( x ) | u | . In what follows, for simplicity, we write k u k ε instead of k u k H A,V,ε .2.2.
Hardy and Kato inequalities.
In dimensions N ≥
3, the Hardy inequality for functions u ∈ D , ( R N , C ) writes (cid:18) N − (cid:19) Z R N | u ( x ) | | x | d x ≤ Z R N |∇ u | , and for functions u ∈ D , A,ε ( R N , C ) ε (cid:18) N − (cid:19) Z R N | u ( x ) | | x | d x ≤ ε Z R N |∇| u || ≤ Z R N | ( iε ∇ + A ) u | , (2.3)for any ε >
0. Furthermore, we recall the following Kato’s inequalities. First, for functions u ∈ L ( R N , C ) with ∇ u ∈ L ( R N , C N ), we definesign( u )( x ) = ¯ u ( x ) | u ( x ) | u ( x ) = 00 u ( x ) = 0 . We have ∆ | u | ≥ Re (sign( u )∆ u ) . (2.4)We also have a similar inequality in presence of a magnetic potential A ∈ L ( R N , R N ), ε ∆ | u | ≥ − Re (cid:16) sign( u )( iε ∇ + A ) u (cid:17) . (2.5)Throughout the text, we will use an auxiliary Hardy type potential. This potential was firstintroduced in [30, 31] to extend the penalization method of del Pino and Felmer to compactlysupported potentials V . For N ≥
3, we define the function H : R N → R by H ( x ) = κ | x | ((log | x | ) + 1) β , for β > < κ < (cid:16) N − (cid:17) . Notice that, for all x ∈ R N , we have H ( x ) ≤ κ | x | , or H ( x ) ≤ κ | x | | log | x || β . (2.6) ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 9
The interest of this auxiliary potential comes mainly from the following comparison principle for − ∆ − H which was proved in [7]. Lemma 2.3.
Let N ≥ and Ω ⊂ R N \ { } be a smooth domain. Let v, w ∈ H loc (Ω , R ) be suchthat ∇ ( w − v ) − ∈ L (Ω) , ( w − v ) − / | x | ∈ L (Ω) and − ∆ w − H ( x ) w ≥ − ∆ v − H ( x ) v, ∀ x ∈ Ω . Moreover, if ∂ Ω = ∅ , assume that w ≥ v on ∂ Ω . Then, w ≥ v in Ω . We also point out that H can be compared to the Hardy potential C/ | x | which is a criticalpotential, both at zero and at infinity. Indeed, if V behaves like 1 / | x | α at infinity, for α ≥ H A,V,ε ( R N , C ) and D , A,ε ( R N , C ). It means that the condition R R N V ( x ) | u | < ∞ is unnecessary in that case. The same holds true if V is singular at 0 and behavesas 1 / | x | α for α ≤ Notations adapted to the cylindrical symmetry.
From now on, we deal with dimension N = 3. As we will work with functions having cylindrical symmetry, that is functions such that u ◦ g = u for g ∈ G , where G is defined in (1 . ρ ∈ R + and x ∈ R ,where ρ = ( x + x ) / , x = ( x , x , x ). The angular variable θ ∈ [0 , π ) plays no role. However,even if those functions only depend on ρ and x , there are still functions defined in R and withsome abuse of notations we will write either u ( ρ, x ) or u ( x , x , x ) depending on the situation.We also recall the distance adapted to cylindrical symmetry already mentioned in the introduc-tion: for y, z ∈ R , d cyl ( y, z ) = (cid:16) ( ρ y − ρ z ) + ( y − z ) (cid:17) / , for ρ y = ( y + y ) / and ρ z = ( z + z ) / , as well as the cylindrical ball B cyl ( x, r ) = n y ∈ R | d cyl ( x, y ) < r o , for r > x ∈ R .The following lemma gives us some compact embedding of the magnetic Sobolev spaces withcylindrical symmetry. Lemma 2.4.
Assume that Ω ⊂ R is an open bounded set such that g (Ω) = Ω for every g ∈ G and < ρ < ρ < ρ for every ( ρ cos θ, ρ sin θ, x ) ∈ Ω . Then, the space n u ∈ H A,ε (Ω , C ) | u ◦ g = u for every g ∈ G o is compactly embedded in L q (Ω) , for ≤ q < + ∞ .Proof. First, since Ω is bounded and A ∈ C ( R , R ), we have seen in Lemma 2.2 that this spaceis equivalent to (cid:8) u ∈ H (Ω , C ) | u ◦ g = u for every g ∈ G (cid:9) . Since u depends only on ρ and x , wecan write the square of the H -norm of u as Z Ω (cid:16) |∇ u | + | u | (cid:17) d x d x d x = 2 π Z Ω (cid:16) | ∂ ρ u | + | ∂ x u | + | u | (cid:17) ρ d ρ d x , where Ω is the parametrization of Ω in the ρ, x variables. Now, take a bounded sequence ( u n ) n ⊂ H A,ε (Ω , C ). Considering each u n as a function of the two variables ρ, x in R , we infer that thesequence is bounded as a sequence ( u n ) n ⊂ H (Ω ). We can then use the compact embedding indimension 2 to conclude. (cid:3) The limit problem
Because of the symmetry, our solutions will concentrate on circles and the limit problem will holdin R . The aim of this section is to describe such a limit problem. Consider a constant potential A : R → R and a positive constant a >
0. The equation( i ∇ + A ) u + a u = | u | p − u, y = ( y , y ) ∈ R (3.1)will be referred to as the limit equation associated to the problem (1.1). For solutions concentratingaround a circle of radius ρ >
0, we will have A = ( φ ( ρ , ,
0) and a = c ( ρ , + V ( ρ , . By lemma 2.2, the weak solutions of (3.1) are critical points of the functional J A a : H ( R , C ) → R defined by J A a ( u ) = 12 Z R h | ( i ∇ + A ) u | + a | u | i d y − p Z R | u | p d y. (3.2)Any nontrivial critical point u ∈ H ( R , C ) of J A a belongs to the Nehari manifold N A a = n u ∈ H ( R , C ) | u h ( J A a ) ′ ( u ) , u i = 0 o . A solution u ∈ H ( R , C ) is called a least energy solution, or ground state, of (3.1) if J A a ( u ) = inf v ∈N A a J A a ( v ) . The following lemma states that any least energy solution of the limit problem (3.1) is real upto a change of gauge and a complex phase.
Lemma 3.1.
Suppose v is a least energy solution of equation (3.1) . Then v ( y ) = w ( y − y ) e iα e iA · y , for some α ∈ R , y ∈ R and where w is the unique radially symmetric real positive solution of thescalar equation − ∆ w + a w = | w | p − w in R . (3.3) Proof.
First, we consider the functional J a : H ( R , C ) → R associated to equation (3.3) J a ( u ) = 12 Z R |∇ u | + a | u | d y − p Z R | u | p d y. Again, any nontrivial critical point u ∈ H ( R , C ) of J a belongs to the Nehari manifold N a . Byperforming the change of gauge v ( y ) = e iA · y u ( y ) (3.4)on functions v ∈ N A a and u ∈ N a , we observe that there is an isomorphism between the two Neharimanifolds. Indeed, any least energy solution v of J A a provides a least energy solution u of J a by(3.4) and vice-versa.Since it is well-known, see for example [25, Lemma 7], that the set of complex valued least energysolutions u of J a can be written as { u ( y ) = e iα w ( y − y ) , α ∈ R , y ∈ R } , the proof is completed. (cid:3) ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 11
We may now define the ground energy function E : R × R + \{ } → R + by E ( A , a ) = inf v ∈N A a J A a ( v ) . The following lemma gives some properties of this ground energy function. We refer to [7] or [34]for more details.
Lemma 3.2.
For every ( A , a ) ∈ R × R + \{ } , E ( A , a ) is a critical value of J A a and we havethe following variational characterization E (0 , a ) = E ( A , a ) = inf v ∈ H ( R , C ) \{ } max t ≥ J A a ( v ) . Moreover, (i) for every A ∈ R , a ∈ R + \{ } 7→ E ( A , a ) is continuous; (ii) for every A ∈ R , a ∈ R + \{ } 7→ E ( A , a ) is strictly increasing.In fact, for our nonlinearity E (0 , a ) = E ( A , a ) = E ( A a − p − , a p − = E (0 , a p − . (3.5)Finally, the concentration function M : R + × R + → R + , already introduced in (1.6), is definedmore precisely by M ( ρ, | x | ) = 2 πρ E (0 , c ( ρ, | x | ) + V ( ρ, | x | )) = 2 πρ h c ( ρ, | x | ) + V ( ρ, | x | ) i p − E (0 , . We will look for solutions concentrating around local minima of M .4. The penalization scheme
The functional associated to equation (1.1) is given by Z R (cid:16) | ( iε ∇ + A ) u | + V | u | (cid:17) d x − p Z R | u | p d x. It is natural to consider this functional in the Sobolev space H A,V,ε ( R , C ). However, the mereassumptions on V , and more particularly the fact that V can decay to zero at infinity, do notensure that H A,V,ε ( R , C ) is embedded in the L p ( R , C ). Then, the last term of the functional isnot necessarily finite. Moreover, even if we assume that V is bounded away from zero, the functionalwould have a mountain-pass geometry in H A,V,ε ( R , C ), but the Palais-Smale condition could failwithout further specific assumptions on V .For those reasons, following del Pino and Felmer [19], we truncate the nonlinear term througha penalization outside the set where the concentration is expected. Basically, the penalizationapproach consists in modifying the nonlinearity outside the bounded set Λ, where Λ verifies (1.7)and (1.8), in the following way ˜ f ( x, s ) = min { µV ( x ) s, f ( s ) } , where 0 < µ <
1. The penalized functional, given by Z R (cid:16) | ( iε ∇ + A ) u | + V | u | (cid:17) d x − Z R ˜ F ( | u | ) d x, where ˜ F ( τ ) = R τ ˜ f ( s ) ds , has the mountain-pass geometry and we recover the Palais-Smale con-dition thanks to the penalization, so that we can easily deduce the existence of a mountain passcritical point u . Then, if we succeed to show that f ( u ) ≤ νV ( x ) u outside the set Λ, we recover asolution of the initial problem.We will argue slightly differently for two reasons. The first one is that this approach works finewhen V stays bounded away from zero, or at least when V does not converge to fast to zero at infinity. We do not want to restrict our assumptions on the potentials V to this class. To solve thisissue, we will add the term ε H ( x ) to V in the modified nonlinearity. This penalization approachwas first introduced in [30, 31] as an extension of [8] and subsequently used in [7]. The secondreason, as already said before, is that our functions are complex-valued. We will then perform thepenalization on the modulus of the unknown.4.1. The penalized functional.
We fix µ ∈ (0 , g ε : R × R + → R by g ε ( x, s ) = χ Λ ( x ) f ( s ) + (1 − χ Λ ( x )) min n ( ε H ( x ) + µV ( x )) , f ( s ) o (4.1)for f ( s ) = s p − . Let G ε ( x, s ) = R s g ε ( x, σ ) d σ . There exists 2 < θ ≤ p such that0 < θG ε ( x, s ) ≤ g ε ( x, s ) s ∀ x ∈ Λ , ∀ s > , (4.2)0 < G ε ( x, s ) ≤ g ε ( x, s ) s ≤ ( ε H ( x ) + µV ( x )) s ∀ x / ∈ Λ , ∀ s > . (4.3)Moreover, we have that g ε ( x, s ) is nondecreasing ∀ x ∈ R , (4.4)which is a useful property, see for example [34].In the following we look for cylindrically symmetric solutions of the penalized equation( iε ∇ + A ) u + V ( x ) u = g ε ( x, | u | ) u, x ∈ R . (4.5)Let us define the penalized functional J ε : H A,V,ε ( R , C ) → R J ε ( u ) = 12 Z R | ( iε ∇ + A ) u | + V ( x ) | u | − Z R G ε ( x, | u | ) , and the space X ε = n u ∈ H A,V,ε ( R , C ) | u ◦ g = u, ∀ g ∈ G o . By the principle of symmetric criticality [33], the critical points of J ε in X ε are weak solutionsof the penalized problem (4.5), having cylindrical symmetry. Thanks to the properties (4.2) and(4.3), the functional has a mountain pass geometry. Indeed, it clearly displays a local minimum at u = 0, while the infimum is −∞ . Standard arguments imply then the existence of a Palais-Smalesequence ( u n ) n ⊂ X ε for J ε , that is J ε ( u n ) ≤ C and J ′ ε ( u n ) → n → ∞ . To secure the existence of a weak solution of (4.5) for every ε >
0, it only remains to prove that J ε satisfies the Palais-Smale condition, i.e. each Palais-Smale sequence possesses a convergentsubsequence. This is our next aim.4.2. The Palais-Smale condition.Lemma 4.1.
For every ε > , every Palais-Smale sequence for J ε in X ε contains a convergentsubsequence.Proof. We proceed in several steps.
ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 13
Step 1.
As usual, the first step of the proof consists in proving that the Palais-Smale sequence( u n ) n is bounded. By using successively the properties of the Palais-Smale sequence ( u n ) n , (4.2),(4.3) and finally the magnetic Hardy inequality (2.3), we infer that12 k u n k ε = J ε ( u n ) + Z R G ε ( x, | u n | ) ≤ C + Z Λ G ε ( x, | u n | ) + Z Λ c G ε ( x, | u n | ) ≤ C + 1 θ Z R g ε ( x, | u n | ) | u n | + (cid:18) − θ (cid:19) Z Λ c g ε ( x, | u n | ) | u n | ≤ C + 1 θ k u n k ε − θ hJ ′ ε ( u n ) , u n i + (cid:18) − θ (cid:19) Z Λ c (cid:16) ε H ( x ) + µV ( x ) (cid:17) | u n | ≤ C + 1 θ k u n k ε + o (1) k u n k ε + (cid:18) − θ (cid:19) µ Z R V ( x ) | u n | + (cid:18) − θ (cid:19) κ Z R | ( iε ∇ + A ) u n | ≤ C + 1 θ k u n k ε + o (1) k u n k ε + (cid:18) − θ (cid:19) max { µ, κ }k u n k ε . Since θ > { µ, κ } <
1, the inequality (cid:18) − θ (cid:19) (1 − max { µ, κ } ) k u n k ε ≤ C + o (1) k u n k ε leads to the conclusion.From Step 1, we deduce the existence of a function u ∈ X ε such that, up to a subsequence stilldenoted in the same way, u n weakly converges to u . Step 2.
In this step, we prove two useful claims aiming to deduce the strong convergence. Wedefine the closed set A λ = B (0 , e λ ) \ B (0 , e − λ ), where λ ≥ Claim 1 - for every δ > , there exists λ δ ≥ such that lim sup n →∞ ε Z R \ A λδ H ( x ) | u n | < δ. (4.6)The inequality (2.6) together with Hardy inequality (2.3) yields Z R \ A λ H ( x ) | u n | ≤ κλ β Z R | ( iε ∇ + A ) u n | . Since ( u n ) n is bounded, we now infer that for every δ >
0, there exists λ δ ≥ Claim 2 - for every δ > , there exists λ δ ≥ (eventually bigger than the previous one) such that lim sup n →∞ Z R \ A λδ V ( x ) | u n | < δ. (4.7)We first define ξ ∈ C ∞ ( R ) such that 0 ≤ ξ ≤ ξ ( s ) = (cid:26) | s | ≤ | s | ≥ η λ ∈ C ∞ ( R , R ) as η λ ( x ) = ξ (cid:18) log | x | λ (cid:19) . Since ( u n ) n is a bounded Palais-Smale sequence and η λ ≤
1, we deduce that hJ ′ ε ( u n ) , η λ u n i = o (1).We then infer that Z R (cid:16) | ( iε ∇ + A ) u n | + V ( x ) | u n | (cid:17) η λ = Z R g ε ( x, | u n | ) | u n | η λ + Re Z R iε ( iε ∇ + A ) u n · ∇ η λ u n + o (1) . (4.8)Since ¯Λ ⊂ R \{ } , there exists λ ≥ ⊂ A λ . Then, if we take λ ≥ λ , we have η λ = 0 on Λ. Now, using (4.3) and the above remark, we get Z R g ε ( x, | u n | ) | u n | η λ = Z Λ c g ε ( x, | u n | ) | u n | η λ (4.9) ≤ Z Λ c (cid:16) ε H ( x ) + µV ( x ) (cid:17) | u n | η λ ≤ Z R (cid:16) ε H ( x ) + µV ( x ) (cid:17) | u n | η λ . Next, using the properties of η λ and the magnetic Hardy inequality (2.3), we deduce thatRe Z R iε ( iε ∇ + A ) u n · ∇ η λ u n ≤ ε (cid:12)(cid:12)(cid:12)(cid:12)Z R ( iε ∇ + A ) u n ) · ∇ η λ u n (cid:12)(cid:12)(cid:12)(cid:12) (4.10) ≤ Cελ (cid:18)Z R | ( iε ∇ + A ) u n | (cid:19) / Z R | u n | | x | ! / ≤ Cλ Z R | ( iε ∇ + A ) u n | . Combining (4.8), (4.9) and (4.10), we get the estimate Z R \ A λ (cid:16) | ( iε ∇ + A ) u n | + (1 − µ ) V ( x ) | u n | (cid:17) ≤ Z R (cid:16) | ( iε ∇ + A ) u n | + (1 − µ ) V ( x ) | u n | (cid:17) η λ ≤ Cλ k u n k ε + ε Z R H ( x ) | u n | η λ + o (1) , for λ ≥ λ . Finally, thanks to (4.6), if we take λ > λ δ , the second term in the right hand sideis smaller than δ . It follows that, for every δ >
0, we can choose (a new) λ δ ≥ { λ , λ δ } suchthat (4.7) holds. Step 3.
We are now in a position to deduce the strong convergence. We compute k u n − u k ε = h J ′ ε ( u n ) , u n − u i − h J ′ ε ( u ) , u n − u i + Re Z R h g ε ( x, | u n | ) u n − g ε ( x, | u | ) u i ( u n − u ) . From Step 1, we know that ( u n ) n is bounded so that in the right hand side, the first two termsconverge to zero. For the last term in the right hand side, we divide the integral in three pieces.We treat separately the integrals on Λ, A λ δ \ Λ and R \ A λ δ and we next prove that they convergeto zero.For the integral on Λ, we can use the fact that u n ∈ H A,ε (Λ , C ) because inf Λ V > u n ∈ X ε .Then, we conclude by using the compact embedding of H A,ε (Λ , C ) in L q (Λ , C ) for 2 ≤ q < + ∞ .Indeed, Lemma 2.4 applies since Λ is bounded away from the axis x .For the integral in A λ δ \ Λ, we use the fact that u n ∈ H ( A λ δ \ Λ , C ). Indeed, A λ δ \ Λ is bounded.In dimension 3, this space is compactly embedded in L q ( A λ δ \ Λ , C ) for 1 ≤ q <
6. Moreover, thepenalization g ε is bounded on this bounded set. This and the strong convergence in L allows toconclude. ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 15
The claims in Step 2 were intended to treat the remaining integral. Indeed, using (4.6) and (4.7),we infer that lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re Z R \ A λδ h g ε ( x, | u n | ) u n − g ε ( x, | u | ) u i ( u n − u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n →∞ Z R \ A λδ (cid:16) ε H ( x ) + µV ( x ) (cid:17) (cid:16) | u n | + | u | (cid:17) ≤ C (1 + µ ) δ. Then, since δ > (cid:3)
As a direct consequence, we deduce the existence of a least energy solution of the penalizedproblem (4.5).
Theorem 4.2.
Let g ε : R × R + → R defined in (4.1) satisfy (4.2) , (4.3) , (4.4) and V ∈ C ( R \{ } ) verify the hypothesis of Section 2.2. Then, for every ε > , the functional J ε has a non trivialcritical point u ε ∈ X ε , which is also a weak solution of (4.5) , characterized by c ε = J ε ( u ε ) = inf u ∈X ε \{ } max t> J ε ( tu ) . (4.11)This solution u ε belongs to W ,q loc ( R \{ } ) for 2 ≤ q < + ∞ and therefore to C ,α loc ( R \{ } ). Wecannot hope a better regularity since the penalization g ε is not even continuous.In the next section, we estimate the critical value c ε from above. In the study of the asymptoticsof the solutions u ε , this upper estimate will be useful to determine that the concentration occursexactly in Λ.4.3. Upper estimate of the mountain pass level.Proposition 4.3 (Upper estimate of the critical value c ε ) . Suppose that the assumptions of Theo-rem 4.2 are satisfied. For every ε > small enough, the critical value c ε defined in (4.11) satisfies lim inf ε → ε − c ε ≤ inf Λ ∩H ⊥ M . (4.12) Moreover, there exists
C > such that the solution u ε found in Theorem 4.2 satisfies k u ε k ε ≤ Cε . (4.13) Proof.
Let x = ( ρ cos θ, ρ sin θ, ∈ Λ ∩ H ⊥ , with ρ > θ ∈ [0 , π ), be such that M ( x ) =inf Λ ∩H ⊥ M . The existence of x is ensured by the continuity of M on Λ and (1.7). Consider thefunctional J A a defined by (3.2), with a = (cid:2) c ( x ) + V ( x ) (cid:3) and A = ( φ ( x ) , η ∈ C ∞ ( R + × R ) such that 0 ≤ η ≤ η = 1 in a small neighbourhood of( ρ , ρ , k∇ η k L ∞ ≤ C . Wedefine the cylindrically symmetric function u ( x , x , x ) = u ( ρ, x ) = η ( ρ, x ) v (cid:18) ρ − ρ ε , x ε (cid:19) , where v is the least-energy solution of J A a . If we perform the change of variables y = ρ − ρ ε and y = x ε in the computation of J ε ( tu ), we get J ε ( tu ) = t Z R h | ( iε ∇ + A ) u | + V ( x ) | u | i d x − Z R G ε ( x, t | u | ) d x = 2 πε t Z ∞− ρ ε Z R { η ( ρ + εy , εy ) h | ( i ∇ + ( φ ( ρ + εy , εy ) , A ( ρ + εy , εy )) v | i + η ( ρ + εy , εy ) h c ( ρ + εy , εy ) + V ( ρ + εy , εy ) i | v | } ( ρ + εy ) d y d y − πε Z ∞− ρ ε Z R G ε ( ρ + εy , εy , t | v | η ) ( ρ + εy ) d y d y + o ( ε ) . The term o ( ε ) includes the terms where the derivatives were applied to η instead of v . This termis controlled thanks the the compactness of the support of η and the control k∇ η k L ∞ ≤ C . Finally,as η is compactly supported around ( ρ , G ε ( x, t | v | η ) will coincide with F ( t | v | η ) for ε smallenough. We then deduce that lim inf ε → ε − J ε ( tu ) ≤ πρ J A a ( tv ) . Now we exploit the fact that c ε is the least-energy level for J ε and v is the least-energy functionfor J A a , as well as Lemma 3.1 where w is the least energy solution to J a , to obtainlim inf ε → ε − c ε ≤ lim inf ε → ε − max t> J ε ( tu ) ≤ πρ max t> J A a ( tv )= 2 πρ J A a ( v ) = 2 πρ J a ( w ) . The last equality follows from (3.5).To deduce the second statement of the proposition, we argue as in Step 1 of the proof of Lemma4.1, with the extra properties that J ′ ε ( u ε ) = 0, because we have the additional information that u ε is a critical point, and J ε ( u ε ) = c ε ≤ Cε . We then infer that (cid:18) − θ (cid:19) (1 − max { κ, µ } ) k u ε k ε ≤ Cε . (cid:3) Asymptotic estimates
In this section, we study the behaviour of solutions when ε →
0. With those estimates at hand,we will be able to prove that the solutions of the penalized problem solve the original equation for ε small enough.5.1. No uniform convergence to on Λ . We start by proving that the solution u ε does notconverge uniformly to 0 in Λ as ε → Proposition 5.1.
Suppose the assumptions of Theorem 4.2 are satisfied and let ( u ε ) ε ⊂ X ε be thesolutions found in Theorem 4.2. Then, lim inf ε → k u ε k L ∞ (Λ) > . Proof.
By contradiction, assume that there exists a sequence ( ε n ) n ⊂ R + such that ε n → k u ε n k L ∞ (Λ) → n → + ∞ . Using Kato inequality (2.5) and the equation (4.5), we obtain − ε n (∆ + H ) | u ε n | + (1 − µ ) V | u ε n | ≤ − ε n H | u ε n | − µV | u ε n | + g ε n ( x, | u ε n | ) | u ε n | . ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 17
By (4.3), we infer that the right hand side of the last inequality is non positive in Λ c . On theother hand, since we assume that k u ε n k L ∞ (Λ) →
0, the facts that p > V ( x ) > | u ε n | p − ≤ µV ( x ) | u ε n | in Λ for n large. We thus conclude that − ε n (∆ + H ( x ) | ) | u ε n | + (1 − µ ) V ( x ) | u ε n | ≤ , in R . We then reach a contradiction because the comparison principle (Lemma 2.3) implies that | u ε n | = 0for large n . (cid:3) Estimates on the rescaled solutions.
As we have seen in Proposition 4.3, the norm of thesolution u ε is of the order ε . It is then natural to rescale u ε around some family of points ( ρ ε , x ,ε )as v ε ( y ) = u ε ( x ε ) = u ε ( ρ ε + εy , x ,ε + εy ) , (5.1)where ( x ε ) ε = ( ρ ε cos θ, ρ ε sin θ, x ,ε ) ε ⊂ ¯Λ, θ ∈ [0 , π [. The rescaled solution is defined for y =( y , y ) ∈ ( − ρ ε /ε, + ∞ ) × R . The following lemma shows the convergence of those rescaled sequencesof solutions. Lemma 5.2 (Convergence of the rescaled solutions) . Suppose the assumptions of Theorem 4.2 aresatisfied. Let ( ε n ) n ⊂ R + and ( x n ) n = ( ρ n cos θ, ρ n sin θ, x ,n ) n ⊂ ¯Λ , θ ∈ [0 , π ) be such that ε n → and x n → ¯ x = (¯ ρ cos θ, ¯ ρ sin θ, ¯ x ) ∈ Λ , as n → + ∞ . Set ¯ A = ( φ (¯ ρ, ¯ x ) , A (¯ ρ, ¯ x )) , ¯ a = c (¯ ρ, ¯ x ) + V (¯ ρ, ¯ x ) . Consider the sequence of solutions ( u ε n ) n ⊂ X ε n found in Theorem 4.2. There exists v ∈ H ( R , C ) such that, up to a subsequence, v ε n → v in C ,α loc ( R , C ) for α ∈ (0 , , where ( v ε n ) n is the sequence defined by (5.1) , v solves the equation ( i ∇ + ¯ A ) v + ¯ av = ¯ g ( y, | v | ) v in R , (5.2) with ¯ g ( y, | v | ) = χ ( y ) f ( | v | ) + (1 − χ ( y )) min { µV (¯ ρ, ¯ x ) , f ( | v | ) } , (5.3) χ being the limit a.e. of χ n ( y ) = χ Λ ( ρ n + ε n y , x ,n + ε n y ) . Moreover, we have π ¯ ρ Z R (cid:16) | ( i ∇ + ¯ A ) v | + ¯ a | v | (cid:17) d y =lim R → + ∞ lim inf n → + ∞ ε − n Z B cyl ( x n ,ε n R ) (cid:16) | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | (cid:17) d x. (5.4) Proof.
We proceed again in several steps.
Step 1: Convergence of the sequence ( v ε n ) n . First, the equation solved by v ε n is thefollowing( i ∇ + A n ) v ε n − ε n ρ n + ε n y ∂v ε n ∂y + i ε n ρ n + ε n y φ n v ε n + h V n + c n i v ε n = g ε n ,n ( y, | v ε n | ) v ε n . (5.5)The two-dimensional magnetic potential A n ( y ) is given by A n ( y ) = ( φ n ( y ) , A ,n ( y )) and the otherfunctions are defined by φ n , A ,n , c n , V n , g ε n ,n ( y ) := φ, A , c, V, g ε n ( ρ n + ε n y , x ,n + ε n y ) . By using the definition of v ε n and (4.13), we obtain the following inequality Z + ∞− ρnεn Z R h | ( i ∇ + A n ) v ε n | + ( V n + c n ) | v ε n | i ( ρ n + ε n y ) d y d y = 12 πε n k u ε n k ε n ≤ C, (5.6) for C > n .Next, we choose e sequence R n such that R n → + ∞ and ε n R n → n → + ∞ , and we definethe cut-off function η R n ∈ C ∞ c ( R ) such that 0 ≤ η R n ≤ η R n ( y ) = (cid:26) | y | ≥ R n | y | ≤ R n / k∇ η R n k L ∞ ≤ C/R n for some C >
0. Since Λ ∩ H = ∅ , we have that ρ n → ¯ ρ >
0, and then, for n sufficiently large, ε n R n < ¯ ρ/ < ρ n . Set w n ( y ) = η R n ( y ) v ε n ( y ), where v ε n is extended by 0 whereit is not defined (anyway η R n = 0 therein).We now estimate the L -norm of | w n | . Observe that if y + y ≤ R n , then ( ρ − ρ n ) +( x − x ,n ) ≤ ε n R n , so that for n large enough, ρ n + ε n y ∈ Λ. Hence, since by hypothesis inf Λ ( c + V ) > ρ n − ε n R n ) > ¯ ρ/ n large enough, we infer that Z R | w n | d y d y ≤ Z B (0 ,R n ) | v ε n | d y d y (5.7) ≤ ρ sup ¯Λ c + V Z B (0 ,R n ) | v ε n | ( c n + V n )( ρ n + ε n y ) d y d y . Using the fact that B (0 , R n ) ⊂ (cid:16) − ρ n ε n , + ∞ (cid:17) × R , for n large enough and (5.6), we deduce theestimate Z R | w n | d y d y ≤ ρ sup ¯Λ c + V Z + ∞− ρnεn Z R | v ε n | ( c n + V n )( ρ n + ε n y ) d y d y ≤ C. Next, we study the L -norm of ∇| w n | . By using the diamagnetic inequality (2.2) and arguing asbefore, we get Z R |∇| w n || d y d y ≤ Z R | ( i ∇ + A n )( η R n v ε n ) | d y d y ≤ Z R | ( i ∇ + A n ) v ε n | η R n d y d y + 2 Z R |∇ η R n | | v ε n | d y d y ≤ ρ sup ¯Λ c + V Z B (0 ,R n ) (cid:16) | ( i ∇ + A n ) v ε n | + ( c n + V n ) | v ε n | (cid:17) ( ρ n + ε n y ) d y d y ≤ C. We have just shown that ( | w n | ) n ⊂ H ( R , R ) is a bounded sequence. Hence, there exists a function | v | ∈ H ( R , R ) such that, up to a subsequence, | w n | converges weakly to | v | . Moreover, we deducefrom Sobolev embeddings that the convergence is strong in L ploc ( R , C ) for 2 ≤ p < + ∞ .To prove the convergence in C ,α loc , we consider any compact set K ⊂ R . For n sufficiently large,we have K ⊂ B (0 , R n ) which implies w n = v n in K . In that compact set K , w n solves the equation(5.5) and using a standard bootstrap argument (see for example [24, Theorem 9.1]) and the factthat w n ∈ L p ( K, C ) for 2 ≤ p < + ∞ , we conclude thatsup n k w n k W ,p ( K ) ≤ C. Finally, since this estimate holds for all 2 ≤ p < + ∞ , Sobolev embeddings imply that w n = v n converges in C ,α ( K ) to v . The claim then follows from a diagonal procedure. Step 2: Limit equation satisfied by v . Since Λ is smooth, the characteristic functionsconverge a.e. to a measurable function 0 ≤ χ ( y ) ≤
1. We therefore obtain equation (5.2) from(5.5) by using the C ,α loc -convergence. Moreover, if ¯ x ∈ Λ, we remark that ¯ g ( y, | v | ) = f ( | v | ), thatis χ ≡ ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 19
Step 3: Proof of the estimate (5.4) . Using the preceding arguments and the C ,αloc -convergence,we havelim inf n → + ∞ ε − n Z B cyl ( x n ,ε n R ) (cid:16) | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | (cid:17) d x = 2 π lim inf n → + ∞ Z B (0 ,R ) h | ( i ∇ + A n ) v ε n | + ( c n + V n ) | v ε n | i ( ρ n + ε n y ) d y d y = 2 π ¯ ρ Z B (0 ,R ) h | ( i ∇ + ¯ A ) v | + ¯ a | v | i d y d y . Finally, we let R go to + ∞ to complete the proof. (cid:3) Next, we examine the contribution of u ε to the action functional in a neighbourhood of a circle.In particular, we derive a lower estimate on the action of u ε which accounts for the number ofcircles around which u ε is non negligible. By combining the next lemmas with the upper estimateon the critical level c ε , we reach the conclusion that u ε concentrates around exactly one circle. Lemma 5.3 (lower bound in a small ball) . Suppose that the assumptions of Theorem 4.2 aresatisfied. Let ( ε n ) n ⊂ R + and ( x n ) n = ( ρ n cos θ, ρ n sin θ, x ,n ) n ⊂ ¯Λ be such that ε n → and x n → ¯ x = (¯ ρ cos θ, ¯ ρ sin θ, ¯ x ) ∈ ¯Λ as n → + ∞ , θ ∈ [0 , π ) . Let ( u ε n ) n ⊂ X ε n be the solutions foundin Theorem 4.2. If lim inf n → + ∞ | u ε n ( x n ) | > , (5.8) then, up to a subsequence, we have lim inf R → + ∞ lim inf n → + ∞ ε − n Z B cyl ( x n ,ε n R )
12 ( | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | ) − G ε n ( x, | u ε n | ) ≥ M (¯ ρ, ¯ x ) . Proof.
We set again v ε n as in (5.1). First, by (5.8), | v (0) | = lim n → + ∞ | v ε n (0) | >
0, then v is notidentically zero. Moreover, we know from Lemma 5.1 that v satisfies the equation (5.2). Thisimplies that v is a critical point of the functional G ¯ A ¯ a : H ( R , C ) → R defined by G ¯ A ¯ a ( u ) = 12 Z R | ( i ∇ + ¯ A ) u | + ¯ a | u | d y − Z R ¯ G ( y, | u | ) d y, ¯ a and ¯ A being defined in Lemma 5.2, and where¯ G ( y, s ) = 12 Z s ¯ g ( y, σ ) d σ. Since ¯ g ( y, | u | ) ≤ f ( | u | ), it follows immediately that G ¯ A ¯ a ( u ) ≥ J ¯ A ¯ a ( u ) . Since v is a critical point of G ¯ A ¯ a and ¯ g satisfies the property (4.4), we have that G ¯ A ¯ a ( v ) = sup t> G ¯ A ¯ a ( tv ) ≥ inf u ∈ H ( R , C ) sup t> G ¯ A ¯ a ( tu ) ≥ inf u ∈ H ( R , C ) sup t> J ¯ A ¯ a ( tu ) = E ( ¯ A, ¯ a ) = E (0 ,
1) ¯ a p − . By using the C ,α loc -convergence of the sequence ( v ε n ) n , we obtain thatlim inf n → + ∞ ε − n Z B cyl ( x n ,ε n R ) (cid:20) (cid:16) | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | (cid:17) − G ε n ( x, | u ε n | ) (cid:21) d x =2 π lim inf n → + ∞ Z B (0 ,R ) (cid:20) (cid:16) | ( i ∇ + A n ) v ε n | + ( V n + c n ) | v ε n | (cid:17) − G ε n ,n ( y, | v ε n | ) (cid:21) ( ρ n + ε n y ) d y =2 π ¯ ρ Z B (0 ,R ) (cid:20) (cid:16) | ( i ∇ + ¯ A ) v | + ¯ a | v | (cid:17) − ¯ G ( y, | v | ) (cid:21) d y. Finally, we let R → + ∞ to conclude. (cid:3) The following lemma estimates what happens outside the small balls where u ε concentrates.In particular we show that the contribution to the action of u ε is nonnegative so that the lowerestimate from the preceding lemma is meaningful. Lemma 5.4 (Inferior bound outside small balls) . Assume that the assumptions of Theorem 4.2are satisfied. Let ( ε n ) n ⊂ R + and ( x in ) n = ( ρ in cos θ, ρ in sin θ, x i ,n ) n ⊂ ¯Λ be such that ε n → and x in → ¯ x i = (¯ ρ i cos θ, ¯ ρ i sin θ, ¯ x i ) ∈ ¯Λ , for ≤ i ≤ M , as n → + ∞ , θ ∈ [0 , π ) . Let ( u ε n ) n ⊂ X ε n bethe solutions found in Theorem 4.2. Then, up to a subsequence, we have lim inf R → + ∞ lim inf n → + ∞ ε − n Z R \B n ( R ) (cid:16) | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | (cid:17) − G ε n ( x, | u ε n | ) ≥ , (5.9) where B n ( R ) = ∪ Mi =1 B cyl ( x in , ε n R ) . (5.10) Proof.
We consider yet another smooth test function η R,ε n such that η R,ε n = 0 on B n ( R/ η R,ε n =1 on R \B n ( R ) and k∇ η R,ε n k L ∞ ≤ C/ ( ε n R ). From (4.2) and (4.3), we infer that Z R \B n ( R ) (cid:20) (cid:16) | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | (cid:17) − G ε n ( x, | u ε n | ) (cid:21) d x ≥ Z R \B n ( R ) (cid:20) (cid:16) | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | (cid:17) − g ε n ( x, | u ε n | ) | u ε n | (cid:21) d x. If we test the equation (4.5) on ( u ε n η R,ε n ), we obtain0 = Z R \B n ( R ) h | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | − g ε n ( x, | u ε n | ) | u ε n | i d x + Z B n ( R ) \B n ( R/ h | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | − g ε n ( x, | u ε n | ) | u ε n | i η R,ε n d x − iε n Z B n ( R ) \B n ( R/ ∇ η R,ε n · ( iε n ∇ + A ) u ε n u ε n d x. Then, to deduce the estimate (5.9), it is enough to estimate the last two integrals in the annularregion A n ( R ) = B n ( R ) \B n ( R/ A n ( R ) is a bounded set havingthe cylindrical symmetry and such that A n ( R ) ∩ H = ∅ , we can use the compact embeddings fromLemma 2.4. Then, we conclude thatlim inf n → + ∞ ε − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z A n ( R ) h | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | − g ε n ( x, | u ε n | ) | u ε n | i η R,ε n d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim inf n → + ∞ ε − n C (cid:16) I n,R + I qn,R (cid:17) , ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 21 where we denoted I n,R = "Z A n ( R ) h | ( iε n + A ) u ε n | + V ( x ) | u ε n | i d x . Next, we estimate the second termlim inf n → + ∞ ε − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε n Z A n ( R ) ( iε n ∇ + A ) u ε n · ∇ η R,ε n u ε n d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim inf n → + ∞ Cε − n R − Z A n ( R ) | ( iε n ∇ + A ) u ε n || u ε n | d x ≤ lim inf n → + ∞ Cε − n R − Z A n ( R ) (cid:16) | ( iε n ∇ + A ) u ε n | + | u ε n | (cid:17) d x ≤ lim inf n → + ∞ Cε − n R − I n,R . Finally, by taking the lim inf R → + ∞ and using relation (5.4), we obtain that both integrals convergeto zero, which concludes the result. (cid:3) The next lemma combines the informations from the two preceding ones and yields a lower boundon the action of u ε as a function of the points in ¯Λ where the solution concentrates. Lemma 5.5 (lower bound on the critical level) . Suppose that the assumptions of Theorem 4.2are satisfied. Let ( ε n ) n ⊂ R + and ( x in ) n = ( ρ in cos θ, ρ in sin θ, x i ,n ) n ⊂ ¯Λ be such that ε n → and x in → ¯ x i = (¯ ρ i cos θ, ¯ ρ i sin θ, x i ) ∈ ¯Λ , for ≤ i ≤ M , as n → + ∞ , θ ∈ [0 , π ) . Let ( u ε n ) n ⊂ X ε n bethe solutions found in Theorem 4.2. Assume that for every ≤ i < j ≤ M , we have lim sup n → + ∞ d cyl ( x in , x jn ) ε n = + ∞ , (5.11) and lim inf n → + ∞ | u ε n ( x in ) | > . Then it holds lim inf n → + ∞ ε − n c ε n ≥ M X i =1 M (¯ ρ i , ¯ x i ) . Proof.
We infer from the previous lemmas that for every δ >
0, there exists R δ > R > R δ lim inf n → + ∞ ε − n Z R \B n ( R ) (cid:20)
12 ( | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | ) − G ε n ( x, | u ε n | ) (cid:21) d x ≥ − δ lim inf n → + ∞ ε − n Z B cyl ( x in ,ε n R ) (cid:20)
12 ( | ( iε n ∇ + A ) u ε n | + V ( x ) | u ε n | ) − G ε n ( x, | u ε n | ) (cid:21) d x ≥ M (¯ ρ i , ¯ x i ) − δ, where B n ( R ) is defined in (5.10). Then, thanks to the hypothesis (5.11), the balls are disjoint.We then decompose ε − n J ( u ε n ) as the sum of the M integrals on each ball B cyl ( x in , ε n R ) and oneintegral in R \B n ( R ). We then havelim inf n → + ∞ ε − n J ( u ε n ) ≥ M X i =1 M (¯ ρ i , ¯ x i ) − ( M + 1) δ. Since δ > (cid:3)
The following proposition is a key result of the proof. It concludes to the existence of a sequenceof maximum points for u ε in ¯Λ and tells us that that sequence of maximum points will in factconverge to the point of infimum of our concentration function M at the interior of Λ. Proposition 5.6.
Suppose that the assumptions of Theorem 4.2 are satisfied. Let ( u ε ) ε ⊂ X ε bethe solutions found in Theorem 4.2 for ε > . Then, there exist ( x ε ) ε = ( ρ ε cos θ, ρ ε sin θ, x ,ε ) ε ⊂ ¯Λ such that lim inf ε → | u ε ( x ε ) | > . (5.12) Moreover, we have (i) lim sup ε → d cyl ( x ε , H ⊥ ) ε < + ∞ , that is x ,ε → ; (ii) lim inf ε → d cyl ( x ε , ∂ Λ) > ; (iii) lim ε → M ( x ε ) = inf Λ ∩H ⊥ M ; (iv) for every δ > , there exists R δ > , such that for every R > R δ there exist ε R > suchthat, for every ε < ε R , | u ε | < δ in Λ \ B cyl ( x ε , εR ) .Proof. First, observe that the existence of a sequence ( x ε ) ε ⊂ ¯Λ of local maximum points of | u ε | in¯Λ follows from the continuity of u ε . The estimate (5.12) holds because we know from Proposition5.1 that u ε does not converge uniformly to zero in ¯Λ. Proof of assertion (i). By contradiction, assume that there exist sequences ( ε n ) n ⊂ R + and( x n ) n ⊂ ¯Λ such that ε n → x n → ¯ x = (¯ ρ cos θ, ¯ ρ sin θ, ¯ x ) ∈ ¯Λ, θ ∈ [0 , π ) (this is alwayspossible because of the compactness of ¯Λ),lim inf n → + ∞ | u ε n ( x n ) | > , and lim sup n → + ∞ d( x n , H ⊥ ) ε n = + ∞ . Let g ref ∈ G be the reflection with respect to H ⊥ . We know that u ε n ◦ g ref = u ε n , so thatlim inf n → + ∞ | u ε n ( g ref ( x n )) | > . Moreover, by our assumption lim n → + ∞ d cyl ( g ref ( x n ) , H ⊥ ) ε n = + ∞ . Therefore, we infer that lim sup n → + ∞ d cyl ( x n , g ref ( x n )) ε n = + ∞ , We can now use Lemma 5.5 to deduce thatlim inf n → + ∞ ε − n c ε n ≥ ( M (¯ x ) + M ( g ref (¯ x ))) ≥ Λ M , whereas we know from (4.12) in Proposition 4.3 thatlim inf n → + ∞ ε − n c ε n ≤ inf Λ ∩H ⊥ M . This yields the inequality 2 inf Λ M ≤ inf Λ ∩H ⊥ M , which is impossible because of the property (1.7) of the set Λ. ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 23
Proof of assertion (ii). Arguing again by contradiction, assume that there exist sequences( ε n ) n ⊂ R + and ( x n ) n ⊂ ¯Λ such that ε n → n → + ∞ | u ε n ( x n ) | > , and lim n → + ∞ d cyl ( x n , ∂ Λ) = 0 , that is x n → ¯ x = (¯ ρ cos θ, ¯ ρ sin θ, ¯ x ) ∈ ∂ Λ, θ ∈ [0 , π ). By assertion ( i ), we also know that ¯ x ∈ H ⊥ .From Lemma 5.5, we have lim inf n → + ∞ ε − n c ε n ≥ M (¯ ρ, ¯ x ) ≥ inf ∂ Λ ∩H ⊥ M , so that (4.12) in Proposition 4.3 impliesinf ∂ Λ ∩H ⊥ M ≤ inf Λ ∩H ⊥ M , which is again a contradiction to (1.7). Proof of assertion (iii). This is also an easy consequence of Proposition 4.3 and Lemma 5.5.Indeed, using also ( i ), we can still assume the existence of a sequence ( x n ) n such that x n convergesto some ¯ x = (¯ ρ cos θ, ¯ ρ sin θ, ∈ ¯Λ ∩ H ⊥ . Then combining Lemma 5.5 and Proposition 4.3, wededuce that M (¯ x ) ≤ lim inf n → + ∞ ε − n c ε n ≤ inf Λ ∩H ⊥ M . Assume ¯ x ∈ ∂ Λ ∩ H ⊥ . Then, by (1.7) and the last inequality, we haveinf Λ ∩H ⊥ M < inf ∂ Λ ∩H ⊥ M ≤ M (¯ x ) ≤ inf Λ ∩H ⊥ M , which is a contradiction. Henceforth, we deduce that ¯ x ∈ Λ ∩ H ⊥ and lim n → + ∞ M ( x n ) = M (¯ x ) =inf Λ ∩H ⊥ M . Proof of assertion (iv). Assume by contradiction the existence of δ > y n ∈ ¯Λsuch that | u ε n ( y n ) | > δ, and lim n → + ∞ d cyl ( x n , y n ) ε n = + ∞ . Up to a subsequence, we know that y n → ¯ y ∈ ¯Λ ∩ H ⊥ , Then, using again Lemma 5.5, Proposition4.3 and (1.7), we obtaininf Λ ∩H ⊥ M ≥ lim inf n → + ∞ ε − n c ε n ≥ ( M (¯ x ) + M (¯ y )) ≥ Λ ∩H ⊥ M , which is impossible. (cid:3) Solutions of the initial problem
All this section is inspired by [7], where they study concentration of solutions around k -spheresfor Laplacian problems. Linear inequation outside small balls.Lemma 6.1.
Suppose that the assumptions of Theorem 4.2 are satisfied. Let ( u ε ) ε ⊂ X ε be thesolutions found in Theorem 4.2. Let ( x ε ) ε ⊂ ¯Λ , found in Proposition 5.6, be such that lim inf ε → | u ε ( x ε ) | > . Then, there exists r > such that for every r > r , there exists ε r > such that for every ε < ε r , − ε (∆ + H ) | u ε | + (1 − µ ) V | u ε | ≤ in R \ B cyl ( x ε , εr ) . Proof.
First, we have that µV ( x ) ≥ δ > , for x ∈ Λ. By Proposition 5.6 (iv), there exists r > r > r there exist ε r > ε < ε r , | u ε ( x ) | p − < δ in Λ \ B cyl ( x ε , εr ) . Then, we use the Kato inequality (2.4) to obtain − ε (∆ + H ) | u ε | + (1 − µ ) V | u ε | ≤ | u ε | p − − µV | u ε | − ε H | u ε | < \ B cyl ( x ε , εr ) . Now, in R \ Λ, we use again the Kato inequality to obtain − ε (∆ + H ) | u ε | + (1 − µ ) V | u ε | ≤ R \ Λ , by the definition of the nonlinearity g ε in R \ Λ. This concludes the proof. (cid:3)
Barrier functions.
Once we can construct functions w ε verifying the opposite inequation − ε (∆ + H ) w ε + (1 − µ ) V w ε ≥ R \ B cyl ( x ε , εr )with some convenient boundary conditions on ∂B cyl ( x ε , εr ), Lemma 6.1 suggests that we can usethe comparison principle to obtain an upper bound on | u ε | . Those functions w ε will be chosen insuch a good way that the bound | u ε | ≤ Cw ε imply that | u ε | p − ≤ µV ( x ) + ε H ( x ) for all x ∈ R \ Λ,so that we recover solutions of the initial problem (1.1).We now define more precisely the notion of barrier functions.
Definition 6.2.
Let ( x ε ) ε ⊂ R and r > . We say that ( w ε ) ε ⊂ C ,α ( R \{ }\ B cyl ( x ε , εr )) is afamily of barrier functions if there exists ε > such that, for every ε < ε , we have that (i) w ε satisfies the inequation − ε (∆ + H ) w ε + (1 − µ ) V w ε ≥ in R \ B cyl ( x ε , εr );(ii) ∇ w ε ∈ L ( R \ B cyl ( x ε , εr )) ; (iii) w ε ≥ on ∂B cyl ( x ε , εr ) . Construction of the comparison functions.
In this section, we recall how to construct somecomparison functions in Λ and in R \ Λ. Those comparison functions will be used to construct thebarrier functions. We first begin by the construction in R \ Λ. Lemma 6.3.
For every ε > , there exists Ψ ε ∈ C ,α loc ( R \{ }\ Λ) such that ( − ε (∆ + H )Ψ ε + (1 − µ ) V Ψ ε = 0 in R \ Λ , Ψ ε = 1 on ∂ Λ , and Z R \ Λ |∇ Ψ ε | + | Ψ ε | | x | < + ∞ . ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 25
We also have the following estimate for every x ∈ R \ (Λ ∪ { } ) and C > < Ψ ε ( x ) ≤ C | x | . (i) If we assume in addition that ( V ∞ ) holds with α = 2 , then, for every ν > and for every R > , with ¯Λ ⊂ B (0 , R ) , there exist C > and ε > such that, for every ε < ε and forevery x ∈ R \ B (0 , R ) , < Ψ ε ( x ) ≤ C | x | ν ;(ii) If we assume that ( V ∞ ) holds with α < , then, for every ν > and for every R > ,with ¯Λ ⊂ B (0 , R ) , there exist C > and ε > such that, for every ε < ε and for every x ∈ R \ B (0 , R ) , < Ψ ε ( x ) ≤ C exp (cid:16) − ν | x | − α (cid:17) ;(iii) If we assume that ( V ) holds with α = 2 , then, for every ν > and for every < r < ,with B (0 , r ) ∩ ¯Λ = ∅ , there exist C > and ε > such that, for every ε < ε and for every x ∈ B (0 , r ) \ { } , < Ψ ε ( x ) ≤ C | x | ν . (iv) If we assume that ( V ) holds with α > , then, for every ν > and for every < r < ,with B (0 , r ) ∩ ¯Λ = ∅ , there exist C > and ε > such that, for every ε < ε and for every x ∈ B (0 , r ) \ { } , < Ψ ε ( x ) ≤ C exp (cid:16) − ν | x | − α (cid:17) . We refer to [7] for the proof.Now, we construct a comparison function inside of Λ.
Lemma 6.4.
Consider r > . Let ( x ε ) ε = ( ρ ε cos θ, ρ ε sin θ, x ,ε ) ε ⊂ Λ , θ ∈ [0 , π ) , and R > besuch that B cyl ( x ε , R ) ⊂ Λ . We define Φ ε ( x ) = cosh (cid:18) λ R − d cyl ( x, x ε ) ε (cid:19) , where λ > is chosen such that inf ¯Λ V > λ (1 − µ ) . Then, there exists ε > such that, for every ε < ε , − ε (∆ + H ) Φ ε + (1 − µ ) V Φ ε ≥ in B cyl ( x ε , R ) \ B cyl ( x ε , εr ) . Proof.
By simple calculation, we obtain that − ε (∆ + H ) Φ ε + (1 − µ ) V Φ ε =( λ + (1 − µ ) V ) Φ ε − ε H Φ ε + ελ ρ − ρ ε ρ d cyl ( x, x ε ) sinh (cid:18) λ R − d cyl ( x, x ε ) ε (cid:19) ≥ , thanks to the assumption on λ and for ε small enough. (cid:3) Thanks to Proposition 5.6 we remark that the assumption B cyl ( x ε , R ) ⊂ Λ is verified if ε is takensufficiently small. From now, we will always consider that ε ∈ R is taken small enough to havethis property.With those two functions Ψ ε and Φ ε , we are ready to construct the barrier functions. Again werefer to [7] for the proof. Lemma 6.5.
Take r > r ( r introduced in Lemma 6.1). Let λ > be as in Lemma 6.4 and ( x ε ) ε be as in Proposition 5.6. Then, there exists ε > and a family ( w ε ) ε ⊂ C ,α loc ( R \{ }\ B cyl ( x ε , εr )) of barrier functions such that for ε < ε < w ε ( x ) ≤ C exp − λε d cyl ( x, x ε )1 + d cyl ( x, x ε ) ! (1 + | x | ) − ∀ x ∈ R \ ( B cyl ( x ε , εr ) ∪ { } ) . Moreover, if we assume that (i) ( V ∞ ) holds with α = 2 , then, for every ν > and for every R > with ¯Λ ⊂ B (0 , R ) , thereexist C > and ε (eventually smaller than the previous one) such that, for all ε < ε , < w ε ( x ) ≤ C exp − λε d cyl ( x, x ε )1 + d cyl ( x, x ε ) ! | x | − ν ∀ x ∈ R \ B (0 , R ); (6.1)(ii) ( V ∞ ) holds with α < , then, for every ν > and for every R > with ¯Λ ⊂ B (0 , R ) , thereexist C > and ε > such that, for all ε < ε , < w ε ( x ) ≤ C exp − λε d cyl ( x, x ε )1 + d cyl ( x, x ε ) ! exp (cid:16) − ν | x | − α (cid:17) ∀ x ∈ R \ B (0 , R ); (6.2)(iii) ( V ) holds with α = 2 , then, for every ν > and for every r < with B (0 , r ) ∩ ¯Λ = ∅ , thereexist C > and ε > such that, for all ε < ε , < w ε ( x ) ≤ C exp − λε d cyl ( x, x ε )1 + d cyl ( x, x ε ) ! | x | ν ∀ x ∈ B (0 , r ) \ { } ; (6.3)(iv) ( V ) holds with α > , then, for every ν > and for every r > with B (0 , r ) ∩ ¯Λ = ∅ , thereexist C > and ε > such that, for all ε < ε , < w ε ( x ) ≤ C exp − λε d cyl ( x, x ε )1 + d cyl ( x, x ε ) ! exp (cid:16) − ν | x | − α (cid:17) ∀ x ∈ B (0 , r ) \ { } . (6.4)6.3. Back to the original equation.
Thanks to Lemmas 6.1 and 6.5, we obtain an upper boundon | u ε | . Proposition 6.6.
Suppose the assumptions of Theorem 4.2 and Proposition 5.6 are satisfied. Let λ > be as in Lemma 6.4, ( x ε ) ε ⊂ ¯Λ be as in Proposition 5.6 and ( u ε ) ε ⊂ X ε be the solutionsfound in Theorem 4.2. Then, there exists C > and ε > such that, for all ε < ε , < | u ε ( x ) | ≤ C exp − λε d cyl ( x, x ε )1 + d cyl ( x, x ε ) ! (1 + | x | ) − ∀ x ∈ R \ { } . (6.5) Moreover, (6.1) - (6.4) hold for | u ε | in place of w ε if we make the same assumptions on V .Proof. By Lemma 6.1, we know that | u ε | is a subsolution in R \ B cyl ( x ε , εr ), for some r > r .Furthermore, thanks to Lemma 5.2, we know that k u ε k L ∞ ( B cyl ( x ε ,εr )) is bounded for ε < ε . Wededuce that | u ε | ≤ k u ε k L ∞ ( B cyl ( x ε ,εr )) on ∂B cyl ( x ε , εr ). With the comparison principle, we concludethat 0 < | u ε ( x ) | ≤ k u ε k L ∞ ( B cyl ( x ε ,εr )) w ε ( x ) ∀ x ∈ R \ ( B cyl ( x ε , εr ) ∪ { } ) . Finally, since | u ε | is bounded in B cyl ( x ε , εr ), we obtain the estimate (6.5) for all x ∈ R . The otherestimates follow by reasoning in the same way. (cid:3) We can now proof the main Theorem.
ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 27
Proof of Theorem 1.1.
It remains us to prove that u ε is in fact a solution of the initial problem(1.1). For this, we need to show that f ( | u ε | ) = | u ε | p − ≤ ε H ( x ) + µV ( x ) ∀ x ∈ R \ Λ . We prove this for example in the case where we make no assumptions on V (then p > | u ε ( x ) | p − ≤ Ce − λε ( p − (1 + | x | ) − ( p − ≤ ε H ( x ) + µV ( x ) , for small ε . The last inequality is verified since we considered p >
4. Indeed, for | x | large, theright-hand side behaves as 1 / (cid:0) | x | log | x | (cid:1) . The left-hand side decays then faster since it behavesas 1 / | x | p − . For | x | small, the left-hand side behaves as a constant while the right-hand side isunbounded. The other cases may be treated in a similar way. (cid:3) Remark 6.7.
In addition of theorem 1.1, we may also prove that estimates (6.1) - (6.4) hold for | u ε | instead of w ε if we make the corresponding assumptions on V . Another class of symmetric solutions
When A is equal to the Lorentz potential, i.e. A = ( − x , x ,
0) or has the slightly more generalform A ( ρ, θ, x ) = c ( ρ )( − sin θ, cos θ, , (7.1)Esteban and Lions have proposed in [22, Section 4.3] the class of solutions u k := C k (cid:18) x + ix ρ (cid:19) k v k , where k ∈ Z , C k ∈ R \ { } and v k are real cylindrically symmetric solutions of an auxiliary problem.One can check easily that the functions u k solve( iε ∇ + A ) u k + V ( ρ, x ) u k = | u k | p − u k , x ∈ R , (7.2)if and only if the v k are real solutions of − ε ∆ v k + (cid:18) kερ + c ( ρ ) (cid:19) + V ( ρ, x ) ! v k = C p − k | v k | p − v k , x ∈ R . (7.3)The limit equation in R has the form − ∆ w k + (cid:16) c ( ρ ) + V ( ρ , x , ) (cid:17) w k = C p − k | w k | p − w k , (7.4)where ( ρ , x , ) is such that the normalized concentration function M ( ρ, x ) = ρ (cid:16) c ( ρ ) + V ( ρ, x ) (cid:17) p − (7.5)is locally minimized at this point.Observe that this reduction to a real valued problem allows us to use directly the arguments from[7] without much modifications. One can then consider several cases according to the properties of c and V . We do not address all these cases in details. We will focus on the special case which forinstance allows to consider a critical frequency. Existence at the critical frequency.
Remember that the potential V stands for U − E ,where U is the electrical potential and E is the frequency of the standing wave ψ ( x, t ) = e − i E ~ t u ( x ).When E = inf R N U ( x ), we say that E is the critical frequency. When A = 0, the critical frequencywas studied by many authors, starting with the contribution of Byeon and Wang [9, 10] and followedby many others.Byeon and Wang have shown that there exists a standing wave which is trapped in a neighbour-hood of the isolated minimum points of V and whose amplitude goes to 0 as ~ →
0. Moreover,depending upon the local behaviour of the potential function V near the minimum points, thelimiting profile of the standing-wave solutions was shown to exhibit quite different characteristicfeatures. This is in striking contrast with the non-critical frequency case (inf U ( x ) > E ) where thesolution develops a spike in the semiclassical limit.Here we show that even if the frequency is critical, the presence of an external magnetic fieldallows for the existence of a solution concentrating on a circle and whose amplitude does not vanishin the semiclassical limit so that this solution is a spike type solution.Let p > k ∈ Z . Let V ∈ C ( R \{ } ) be nonnegative and such that V ( gx ) = V ( x ) for every g ∈ G . Assume A ∈ C ( R , R ) is of the form (7.1) and such that c ( ρ ) > ρ andlim inf ρ →∞ c ( ρ ) ρ > . With those assumptions, the assumption lim inf | x |→ + ∞ W ( x ) | x | > W = (cid:18) kερ + c ( ρ ) (cid:19) + V ( ρ, x )and ε small. Moreover this potential is nonnegative everywhere and for every 0 < θ < ε > ρ θ,ε > (cid:18) kερ + c ( ρ ) (cid:19) + V ( ρ, x ) ≥ θc ( ρ ) + V ( ρ, x ) , for ρ ≥ ρ θ,ε . Clearly ρ θ,ε → ε → θ .The proof of the following theorem can be easily recovered from [7] with straightforward modi-fications. Theorem 7.1.
With the above conditions on c , V , k and p , assume there exists a bounded G -invariant smooth set Λ ⊂ R such that (1.7) is satisfied with M now defined by (7.5) and (1.8) is satisfied for θc + V , θ ∈ (0 , . If ε > is small enough, the equation (7.3) hasa solution v k,ε such that v k,ε ( gx ) = v k,ε ( x ) for all g ∈ G , v k,ε attains its maximum at some x k,ε = ( ρ k,ε cos θ, ρ k,ε sin θ, x ,k,ε ) ∈ Λ such that (ii) lim inf ε → | v k,ε ( x k,ε ) | > ε → M ( x k,ε ) = inf Λ ∩H ⊥ M ; (iv) lim sup ε → d cyl ( x k,ε , H ⊥ ) ε < + ∞ , that is x ,k,ε → ; (v) lim inf ε → d cyl ( x k,ε , ∂ Λ) > .Finally, there exists C k ∈ R \ { } such that u k,ε = C k (cid:18) x + ix ρ (cid:19) k v k,ε solves (7.2) and for every ν > , the asymptotic estimate < | u k,ε ( x ) | ≤ C exp − λε d cyl ( x, x k,ε )1 + d cyl ( x, x k,ε ) ! | x | − ν ∀ x ∈ R \ { } . ONCENTRATION DRIVEN BY AN EXTERNAL MAGNETIC FIELD 29
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Denis Bonheure,Département de Mathématique, Université libre de Bruxelles,CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgiumand INRIA - Team MEPHYSTO.
E-mail address : [email protected] Silvia Cingolani,Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari,Via E. Orabona 4, 70125 Bari, Italy.
E-mail address : [email protected] Manon NysFonds National de la Recherche Scientifique- FNRS.Département de Mathématique, Université Libre de Bruxelles,CP 214, Boulevard du triomphe, B-1050 Bruxelles, Belgium.Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca,via Bicocca degli Arcimboldi 8, 20126 Milano, Italy.
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