Strongly localized semiclassical states for nonlinear Dirac equations
aa r X i v : . [ m a t h . A P ] J un Strongly localized semiclassical states fornonlinear Dirac equations
Thomas Bartsch, Tian Xu ∗ June 16, 2020
Abstract
We study semiclassical states of the nonlinear Dirac equation − i ~ ∂ t ψ = ic ~ X k =1 α k ∂ k ψ − mc βψ − M ( x ) ψ + f ( | ψ | ) ψ, t ∈ R , x ∈ R , where V is a bounded continuous potential function and the nonlinear term f ( | ψ | ) ψ is superlinear, possibly of critical growth. Our main result deals with standing wavesolutions that concentrate near a critical point of the potential. Standard methodsapplicable to nonlinear Schr¨odinger equations, like Lyapunov-Schmidt reduction orpenalization, do not work, not even for the homogeneous nonlinearity f ( s ) = s p .We develop a variational method for the strongly indefinite functional associated tothe problem. Keywords.
Dirac equation, semiclassical states, standing waves, concentration, stronglyindefinite functional
Standing wave solutions for the nonlinear Schr¨odinger equation − i ~ ∂ t ψ = − ∆ ψ + V ( x ) ψ + f ( | ψ | ) ψ a non-relativistic wave equation, have been in the focus of nonlinear analysis since decades.In particular, semiclassical states that concentrate near a critical point of the potential V ∗ Supported by the National Science Foundation of China (NSFC 11601370, 11771325) and the Alexan-der von Humboldt Foundation of Germany
Mathematics Subject Classification (2010):
Primary 35Q40; Secondary 49J35 | ψ | ψ in one-dimension.Much less is known for the nonlinear Dirac equation − i ~ ∂ t ψ = ic ~ X k =1 α k ∂ k ψ − mc βψ − M ( x ) ψ + f ( x, | ψ | ) ψ, t ∈ R , x ∈ R , a relativistic wave equation and a spinor generalization of the nonlinear Schr¨odinger equa-tion, not even in the case of f being a pure power with subcritical nonlinearity. Here ψ ( t, x ) ∈ C , c is the speed of light, ~ is Planck’s constant, m is the mass of the particleand α , α , α and β are the × complex Pauli matrices: β = (cid:18) I − I (cid:19) , α k = (cid:18) σ k σ k (cid:19) , k = 1 , , , with σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) . The external field M ( x ) represents an arbitrary electric potential depending only upon x ∈ R . The nonlinear coupling f ( x, | ψ | ) ψ describes a self-interaction. Typical examplesfor nonlinear couplings can be found in the self-interacting scalar theories; see [22, 23,31] and more recently [7, 20, 21, 25, 26, 37, 43]. Usually, in Quantum electrodynamicsnonlinear Dirac equations have to satisfy symmetry constraints, in particular the Poincar´ecovariance. Nonlinear Dirac equations modeling Bose-Einstein condensates break thissymmetry, and often the nonlinearity is a power-type function that depends only on thelocal condensate density (see [28–30] for more background from physics).The ansatz ψ ( t, x ) = e iωt/ ~ u ( x ) for a standing wave solution and a change of notation(in particular ε instead of ~ ) leads to an equation of the form − iε X k =1 α k ∂ k u + aβu + V ( x ) u = f ( x, | u | ) u, u ∈ H ( R , C ) . (1.1)This type of particle-like solution does not change its shape as it evolves in time, hence hasa soliton-like behavior. In this paper we investigate the existence of semiclassical states,i.e. solutions u ε of (1.1) for small ε > , that concentrate as ε → at a critical point x ofthe potential V . There are many results of this type for nonlinear Schr¨odinger equations − ε ∆ u + V ( x ) u = g ( u ) , u ∈ H ( R N ) , (1.2)beginning with the pioneering work by Floer and Weinstein [24] and then continued byOh [38, 39] and many others, e.g. [2–6, 8–11, 40]. It has been proved that there existsa family of semiclassical solutions to (1.2) for small ε which concentrate around stablecritical points of the potential V as ε → . The proofs are based on Lyapunov-Schmidttype methods, penalization, and variational techniques.Very few results are available for the nonlinear Dirac equation (1.1) compared withthe nonlinear Schr¨odinger equation. A major difference between nonlinear Schr¨odingerand Dirac equations is that the Dirac operator is strongly indefinite in the sense that boththe negative and positive parts of the spectrum are unbounded and consist of essentialspectrum. It follows that the quadratic part of the energy functional associated to (1.1) hasno longer a positive sign, moreover, the Morse index and co-index at any critical point ofthe energy functional are infinite.In order to compare our result with the existing literature we first present in short thestate of the art. The first result for concentration behavior of the nonlinear Dirac equation(1.1) is due to Ding [13], who considered the case V ≡ and f ( x, | u | ) = P ( x ) | u | p − with p ∈ (2 , subcritical, inf P > , and lim sup | x |→∞ P ( x ) < max P . He obtained aleast energy solution u ε for ε > small that concentrates around a global maximum of P as ε → . A similar result has been obtained in [14] where f = f ( | u | ) is subcriticaland V satisfies a < min V < lim inf | x |→∞ V ( x ) ≤ | V | ∞ < a . Here the solutions u ε concentrate at a global minimum of V . In both papers [13, 14] the solutions are obtainedvia a mountain pass argument applied to a reduced functional. In [18, 44] the authorsconsidered the case of a local minimum of V using a penalization approach analogous tothe one in [10, 11].All papers mentioned so far consider a subcritical nonlinearity f . The only papersdealing with a critical nonlinearity, i.e. where f ( t ) grows as t for t → ∞ , are [15, 16].Both papers assume, in addition to various technical conditions, that V has a global min-imum. The least energy solution is obtained again via a mountain pass argument appliedto a reduced functional. It is essential that the mountain pass level is below the thresholdlevel where the Palais-Smale condition fails. In [15] the authors were also able to obtainsolutions with energy above the mountain pass level using the oddness of the equationand Lusternik-Schnirelmann type arguments, but again the energy levels of the solutionsare below the level where the Palais-Smale condition fails.The distinct new feature of our result is that we find solutions of − iε X k =1 α k ∂ k u + aβu + V ( x ) u = f ( | u | ) u (1.3)localized near a critical point of V that is not necessarily a (local or global) minimum of V . The model nonlinearity we consider is f ( t ) = κt + λt p − with κ, λ > and p ∈ (2 , .We can deal with local minima, local maxima, or saddle points of V , both in the critical( κ > ) and subcritical ( κ = 0 ) case. As a consequence, a least energy solution may notexist, and in the variational setting there is no threshold value below which the Palais-Smale condition holds, so that the methods from [15, 16] do not apply. We have to workat energy levels where the Palais-Smale condition fails which, in the critical case κ > ,leads to a subtle interplay between κ, V, λ, p . Our results are new even in the subcriticalcase where so far only local minima of V have been treated. They are of course new inthe critical case where only global minima of V have been considered.The paper is organized as follows. In the next section we state and discuss our maintheorem. After collecting some basic results on the Dirac operator in Section 3 we inves-tigate the family of equations − i X k =1 α k ∂ k u + aβu + V ( ξ ) u = f ( | u | ) u (1.4)parametrized by ξ ∈ R which appear as limit equations. This will be done in Section 4.In Section 5 we introduce a truncated problem, set up the variational structure, and provethe Palais-Smale condition for the truncated functional in a certain parameter range. Thenin Section 6 we develop a min-max scheme that can be applied to the truncated problem.The proof of a key result, Proposition 6.4, that is needed for the passage to the limit ε → will be presented in Section 7. The delicate analysis in Section 7 is not needed in the caseof a local minimum of V because in that case the lower bound estimate of Proposition 6.4is automatically satisfied. In the final Section 8 we show that the solutions of the truncatedproblem are actually solutions of (1.1) for ε > small enough, thus finishing the proof ofthe main theorem. The proof of a technical lemma will be presented in the Appendix. We set α · ∇ := P k =1 α k ∂ k so that equation (1.3) reads as − iεα · ∇ u + aβu + V ( x ) u = f ( | u | ) u, u ∈ H ( R , C ) . Throughout the paper, we fix the constant a > and assume that the potential V satisfies ( V V ∈ C , ( R ) ∩ L ∞ ( R ) and | V | ∞ < a .Here we use the notation | · | p for the various L p -norms. We also require one of the fol-lowing hypotheses: ( V V is C in a neighborhood of , and is an isolated local maximum or minimum of V . ( V V is C in a neighborhood of , is an isolated critical point, and there exists avector space X ⊂ R such that: ( a ) V | X has a strict local maximum at ; ( b ) V | X ⊥ has a strict local minimum at .In the case of ( V we may assume that { } 6 = X = R so that is a possibly degeneratesaddle point of V .The domain of the quadratic form associated to the Dirac operator is H ( R , C ) .This space embeds into the corresponding L q -spaces for ≤ q ≤ , and the embeddingis locally compact precisely if q < . Therefore the nonlinearity f ( | u | ) u has subcriticalgrowth if f ( s ) s ∼ s p − with < p < , and it has critical growth if p = 3 . In (3.8) belowwe define for λ > , p ∈ (2 , a constant ¯ κ ( V, λ, p ) > that appears in the followingassumptions when the nonlinearity is critical. Here F ( s ) := R s f ( t ) t dt is the primitive of f ( s ) s . ( f f ∈ C [0 , ∞ ) ∩ C (0 , ∞ ) satisfies f (0) = 0 and f ′ ( s ) > for s > . ( f There exist λ > , p ∈ (2 , , κ ∈ [0 , ¯ κ ) with ¯ κ = ¯ κ ( V, λ, p ) defined in (3.8) suchthat f ( s ) ≥ κs + λs p − for s > , and f ′ ( s ) → κ as s → ∞ . ( f There exists θ > such that < θF ( s ) ≤ f ( s ) s + θ − κs for s > .These conditions imply that s f ( s ) s is strictly increasing and superlinear. Condition ( f is a weakened Ambrosetti-Rabinowitz condition. If κ > then the nonlinearity hascritical growth. Theorem 2.1.
Assume that V satisfies ( V and one of ( V or ( V . Suppose that f satisfies ( f , ( f and ( f . Then (1.1) has a solution u ε for ε > small. These solutionshave the following properties. ( i ) | u ε | possesses a global maximum point x ε ∈ R such that x ε → as ε → , and | u ε ( x ) | ≤ C exp (cid:16) − cε | x − x ε | (cid:17) with C, c > independent of ε . ( ii ) The rescaled function U ε ( x ) = u ε ( εx + x ε ) converges as ε → uniformly to a leastenergy solution U : R → C of − iα · ∇ U + aβU + V (0) U = f ( | U | ) U. Remark 2.2.
Thus in the subcritical case κ = 0 equation (1.1) always has solutions withshape as in ( i ) and ( ii ) . We do allow critical growth but the factor κ cannot be too large.The constant ¯ κ depends on | V | ∞ , sup V , λ and p . It is bounded away from by a positivenumber provided V is bounded away from − a and sup V ≤ . Moreover ¯ κ → as | V | ∞ → a . It is an interesting open problem whether the restriction on κ can be removed. We write L q = L q ( R , C ) for q ≥ and H s = H s ( R , C ) for s > . Let D a = − iα · ∇ + aβ denote the self-adjoint operator on L with domain D ( D a ) = H . It is wellknown that the spectrum of D a is purely continuous and σ ( D a ) = σ c ( D a ) = R \ ( − a, a ) .Therefore L possesses the orthogonal decomposition L = L + ⊕ L − , u = u + + u − , (3.1)so that D a is positive definite (resp. negative definite) in L + (resp. L − ). Now let E := D ( | D a | / ) be the form domain of D a endowed with the inner product h u, v i = Re (cid:0) | D a | / u, | D a | / v (cid:1) and induced norm k · k ; here ( · , · ) denotes the L -inner product. This norm is equivalentto the usual H / -norm, hence E embeds continuously into L q for all q ∈ [2 , andcompactly into L qloc for all q ∈ [2 , . Clearly E possesses the decomposition E = E + ⊕ E − with E ± = E ∩ L ± , (3.2)orthogonal with respect to both ( · , · ) and h· , ·i . Since σ ( D a ) = R \ ( − a, a ) , one has a | u | ≤ k u k for all u ∈ E. (3.3)The decomposition of E induces also a natural decomposition of L q for every q ∈ (1 , ∞ ) as proved in [18]. Proposition 3.1.
Setting E ± q := E ± ∩ L q for q ∈ (1 , ∞ ) there holds L q = cl q E + q ⊕ cl q E − q with cl q denoting the closure in L q . More precisely, for every q ∈ (1 , ∞ ) there exists d q > such that d q | u ± | q ≤ | u | q for all u ∈ E ∩ L q . Moreover, the decomposition is invariant when taking derivatives.
Proposition 3.2.
For u ∈ H we have ∂ k u ± = ( ∂ k u ) ± .Proof. The Fourier transformation of D a is given by ( D a u )ˆ( ξ ) = (cid:18) P k =1 ξ k σ k P k =1 ξ k σ k (cid:19) ˆ u + (cid:18) a − a (cid:19) ˆ u, where ˆ u , a C -valued function, denotes the Fourier transform of u ∈ L . It has beenproved in [18] that the Fourier transforms of the orthogonal projections P ± : L → L ± are given by ( P + u )ˆ( ξ ) = (cid:16)
12 + a p a + | ξ | (cid:17) (cid:18) I Σ( ξ )Σ( ξ ) A ( ξ ) (cid:19) ˆ u and ( P − u )ˆ( ξ ) = (cid:16)
12 + a p a + | ξ | (cid:17) (cid:18) A ( ξ ) − Σ( ξ ) − Σ( ξ ) I (cid:19) ˆ u with I being the × identity matrix and A ( ξ ) = p a + | ξ | − aa + p a + | ξ | · I, Σ( ξ ) = X k =1 ξ k σ k a + p a + | ξ | . The proposition follows from the fact that these matrix operations commute with themultiplication by iξ k for k = 1 , , .The proof of our main results will be achieved by variational methods applied to func-tionals J : E → R of the form J ( u ) = 12 (cid:0) k u + k − k u − k (cid:1) + 12 Z R W ( x ) | u | dx − Z R G ( x, | u | ) dx. (3.4)The following reduction process will be very useful. Theorem 3.3.
Let W ∈ L ∞ satisfy | W | ∞ < a and suppose G : R × R +0 → R has theform G ( x, s ) = R s g ( x, t ) tdt where g is measurable in x ∈ R , of class C in s ∈ R +0 andsatisfies(i) ≤ g ( x, s ) s for all x ∈ R ;(ii) g ( x, s ) s = o ( s ) as s → uniformly in x ∈ R ;(iii) ≤ ∂ s (cid:0) g ( x, s ) s (cid:1) ≤ Cs for all x ∈ R , s > , some C > .Then the following hold for J as in (3.4) .a) There exists a C -map h J : E + → E − such that for v ∈ E + and w ∈ E − DJ ( v + w )[ φ ] = 0 for all φ ∈ E − ⇐⇒ w = h J ( v ) and k h J ( v ) k ≤ | W | ∞ a − | W | ∞ k v k + 2 aa − | W | ∞ Z R G ( x, | v | ) dx. b) Setting J red : E + → R , J red ( v ) := J ( v + h J ( v )) , the sets M + ( J ) := { v ∈ E + \ { } : DJ red ( v )[ v ] = 0 } and M ( J ) := { v + h J ( v ) ∈ E \{ } : v ∈ M + ( J ) } = { u ∈ E \{ } : DJ ( u ) | R u ⊕ E − = 0 } are C -submanifolds of E , diffeomorphic to an open subset of the unit sphere SE + = { v ∈ E + : k v k = 1 } .c) If ( v n ) n is a Palais-Smale sequence for J red then { v n + h J ( v n ) } n is a Palais-Smalesequence for J .d) If | g ( x, s ) | = O ( | s | p − ) as | s | → ∞ for some p ∈ (2 , then h J is weakly sequen-tially continuous. The proof of Theorem 3.3 is standard. We refer the reader to [1, 18, 42] for this kindof results. The diffeomorphisms to an open subset of SE + are simply given by u u + k u + k . In the case W ≡ ν ∈ ( − a, a ) the manifold M ( J ) is the Nehari-Pankov manifoldassociated to J . It will be useful that the decomposition E = E + ⊕ E − is independent of W and does not necessarily correspond to the decomposition of E into the positive andnegative eigenspaces associated to D J (0) = P + − P − + W ( x ) . We call J red the reducedfunctional, h J the reduction map, and ( J red , h J ) the reduction couple of J . Remark 3.4.
In the setting of Theorem 3.3, for each v ∈ SE + the map ϕ v ( t ) = J red ( tv ) is C and has at most one critical point t v > , which is a nondegenerate maximum. Thereholds: M + ( J ) = { t v v : v ∈ SE + , ϕ ′ v ( t v ) = 0 } . If G grows super-quadratically in t as t → ∞ then J ( tu ) → −∞ as t → ∞ and ϕ v ( t ) has a unique maximum for each v ∈ SE + . Then M ( J ) and M + ( J ) are diffeomorphicto SE + . It is clear that M ( J ) contains all nontrivial critical points of J , and that for u ∈ E \ { } there holds: DJ ( u ) = 0 ⇐⇒ u − = h J ( u + ) and DJ red ( u + ) = 0 Finally, the infimum of J on M ( J ) can be described as follows: γ ( J ) := inf u ∈M ( J ) J ( u ) = inf v ∈ E + \{ } sup u ∈ R v ⊕ E − J ( u )= inf v ∈ E + \{ } max t> J red ( tv ) = inf v ∈M + ( J ) J red ( v ) (3.5)If γ ( J ) is achieved then it is the ground state energy.Theorem 3.3 applies in particular to the following functional which depends on theparameters ~µ = ( κ, λ, ν, p ) with κ, λ ≥ , | ν | < a and p ∈ (2 , : J ~µ ( u ) = 12 (cid:0) k u + k − k u − k (cid:1) + ν | u | − λp | u | pp − κ | u | . (3.6)In order to define the constant ¯ κ from Theorem 2.1 let S := inf = u ∈ H |∇ u | | u | (3.7)be the best constant for the embedding H ( R , C ) ֒ → L ( R , C ) . Then we define for V as in ( V , λ > , p ∈ (2 , as in ( f , ( f : ¯ κ := (cid:18) a − | V | ∞ a (cid:19) S (cid:0) γ ( J ~µ V ) (cid:1) − with ~µ V := (0 , λ, sup V, p ) . (3.8)The following technical result will be needed later. Lemma 3.5.
For v ∈ E + \ { } the function H ( t ) = I ( tv ) − t I ′ ( tv )[ v ] is of class C andsatisfies H ′ ( t ) > for all t > .Proof. We set ϕ v ( t ) = I ( tv ) so that H ( t ) = ϕ v ( t ) − t ϕ ′ v ( t ) . Since H ′ ( t ) = 12 ϕ ′ v ( t ) − t ϕ ′′ v ( t ) = 12 t (cid:2) ϕ ′ tv (1) − ϕ ′′ tv (1) (cid:3) , it is sufficient to check that ϕ ′ v (1) − ϕ ′′ v (1) > for all v ∈ E + \ { } . Setting K ( u ) = R R G ( x, | u | ) dx , we have by the definition of h J − h h J ( v ) , φ i + Re Z R W ( x )( v + h J ( v )) · φ dx − K ′ ( v + h J ( v ))[ φ ] = 0 (3.9)for all φ ∈ E − . It follows for z v = v + h J ( v ) and w v = h ′ J ( v )[ v ] − h J ( v ) that ϕ ′ v (1) = k v k + Re Z R W ( x ) z v · v dx − K ′ ( z v )[ v ] = J ′ ( z v )[ z v + w v ] . (3.10)Since (3.9) is valid for all v ∈ E + , differentiating yields for all φ ∈ E − : − h h ′ J ( v )[ v ] , φ i + Re Z R W ( x )( v + h ′ J ( v )[ v ]) · φ dx − K ′′ ( v + h J ( v ))[ v + h ′ J ( v )[ v ] , φ ] . φ = h ′ J ( v )[ v ] in the above identity, so that z v + w v = v + φ , we get ϕ ′′ v (1) = k v k + Re Z R W ( x )( v + h ′ J ( v )[ v ]) · v dx − K ′′ ( z v )[ z v + w v , v ]= k v k − k φ k + Z R W ( x ) | v + φ | dx − K ′′ ( z v )[ z v + w v , v + φ ]= J ′′ ( z v )[ z v + w v , z v + w v ]= k v k − k h J ( v ) k + Z R W ( x ) | z v | dx − K ′′ ( z v )[ z v , z v ]+ 2 (cid:18) − h h J ( v ) , w v i + Re Z R W ( x ) z v · w v dx − K ′′ ( z v )[ z v , w v ] (cid:19) + (cid:18) −k w v k + Z R W ( x ) | w v | dx − K ′′ ( z v )[ w v , w v ] (cid:19) = ϕ ′ v (1) + (cid:0) K ′ ( z v )[ z v ] − K ′′ ( z v )[ z v , z v ] (cid:1) + 2 (cid:0) K ′ ( z v )[ w v ] − K ′′ ( z v )[ z v , w v ] (cid:1) − K ′′ ( z v )[ w v , w v ] − k w v k + Z R W ( x ) | w v | dx. Finally we obtain: ϕ ′ v (1) − ϕ ′′ v (1) ≥ Z R G ′ ( x, | z v | ) | w v | + G ′′ ( x, | z v | ) | z v | (cid:16) | z v | + Re z v · w v | z v | (cid:17) dx > For | ν | < a the problem − iα · ∇ u + aβu + νu = f ( | u | ) u, u ∈ E, (4.1)appears as limit equation of (1.1). We begin with the model case − iα · ∇ u + aβu + νu = λ | u | p − u + κ | u | u u ∈ E. and recall the associated energy functional J ~µ from (3.6) with ~µ = ( κ, λ, ν, p ) and κ, λ, p from ( f , ( f . Proposition 4.1.
The infimum γ ( J ~µ ) is attained provided ν satisfies (cid:16) a a − ν − (cid:17) · κ · γ ( J ~µ ) < S , (4.2) where ν − = min { , ν } . Proof.
We only give the proof for κ > since the subcritical case κ = 0 is much easier.Let ( J red~µ , h ~µ ) denote the reduction couple of J ~µ and let ( v n ) n be a minimizing sequencefor J red~µ in M + ( J ~µ ) . Setting u n = v n + h ~µ ( v n ) it is not difficult to check that ( u n ) n isbounded in E , hence it is either vanishing or non-vanishing up to a subsequence (see [34]).If ( u n ) n has a non-vanishing subsequence then we are done, so let us assume to thecontrary that ( u n ) n is vanishing, hence | u n | p → . We first show that this implies γ ( J ~µ ) ≥ γ ( J ~µ ) where ~µ = ( κ, , ν, p ) . (4.3)In order to see this let t n > be defined by t n v n ∈ M + ( J ~µ ) . Observe that k v n k isbounded away from and the nonlinearity in J ~µ is super-quadratic, so that ( t n ) n isbounded. Theorem 3.3 d) now implies | h ~µ ( t n v n ) | p → where h ~µ is the reduction mapfor J ~µ . Now (4.3) follows from γ ( J ~µ ) ≤ J ~µ ( t n v n + h ~µ ( t n v n ))= J ~µ ( t n v n + h ~µ ( t n v n )) + o n (1) ≤ J red~µ ( v n ) + o n (1) = γ ( J ~µ ) + o n (1) . Next we show that J red~µ ( v ) ≥ κ (cid:18) k v k + ν | v | | v | (cid:19) for all v ∈ M + ( J ~µ ) . (4.4)For this we consider the functional I : E \ { } → R , u
7→ k u + k − k u − k + ν | u | | u | . For any v ∈ E + it is easy to see by a direct argument that sup w ∈ E − I ( v + w ) > isachieved by some w v ∈ E − . Moreover, for any c > the set { w ∈ E − : I ( v + w ) ≥ c } is strictly convex because w
7→ k v k − k w k + ν | v + w | − c | v + w | is strictly concave on E − . This also uses | ν | < a . Hence w v is the unique critical point of w I ( v + w ) . On the other hand, for v ∈ M + ( J ~µ ) , we have DJ red~µ ( v )[ v ] = k v k − k h ~µ ( v ) k + ν | v + h ~µ ( v ) | − κ | v + h ~µ ( v ) | , (4.5)hence J red~µ ( v ) = J red~µ ( v ) − DJ red~µ ( v )[ v ] = κ | v + h ~µ ( v ) | . A direct calculation gives DI (cid:0) v + h ~µ ( v ) (cid:1)(cid:12)(cid:12) E − = 0 and I (cid:0) v + h ~µ ( v ) (cid:1) > h ~µ ( v ) = w v . Now (4.4) follows, using (4.5) once more: J red~µ ( v ) = κ | v + h ~µ ( v ) | = 16 κ I ( v + h ~µ ( v )) ≥ κ I ( v ) Finally, the proposition follows from (4.3), (4.4) and k v k + ν | v | | v | ≥ (cid:16) a − ν − a (cid:17) S for all v ∈ M + ( J ~µ ) . (4.6)For the proof of (4.6) we pass to the Fourier domain and recall from [18] that k u k = Z R ( a + | ξ | ) | ˆ u | dξ for all u ∈ E. Since | ν | < a we have ( a + t ) + ν ≥ (cid:18) a − ν − a (cid:19) | t | for all t ∈ R which implies for v ∈ E + \ { } : k v k + ν | v | | v | = R R [( a + | ξ | ) + ν ] · | ˆ v | dξ | v | ≥ (cid:18) a − ν − a (cid:19) R R | ξ || ˆ u | dξ | u | ≥ (cid:18) a − ν − a (cid:19) S Here the last inequality follows from R R | ξ | | ˆ u | dξ | u | = | c ∇ u | | u | = |∇ u | | u | ≥ S for all u ∈ H ( R , C ) and the Calder´on-Lions interpolation theorem (see [41]).Now we consider the energy functional I ν : E → R associated to (4.1) given by I ν ( u ) = 12 (cid:0) k u + k − k u − k (cid:1) + ν | u | − Z R F ( | u | ) dx. (4.7)The hypotheses ( f − ( f imply that I ν satisfies the assumptions of Theorem 3.3 for | ν | < a . Lemma 4.2. If ν ∈ ( − a, a ) satisfies (4.2) then γ ( I ν ) is achieved for all ν ∈ ( − a, ν ] .Moreover, the map ν γ ( I ν ) is continuous and strictly increasing.Proof. For ν ∈ ( − a, ν ] ⊂ ( − a, a ) assumption ( f implies I ν ≤ J ~µ ≤ J ~µ , where ~µ = ( κ, λ, ν , p ) and ~µ = (0 , λ, ν , p ) . It follows that γ ( I ν ) ≤ γ ( J ~µ ) ≤ γ ( J ~µ ) . A3similar argument as in the proof of Proposition 4.1 implies the existence of a nontrivialcritical point u ν for I ν such that u + ν is the minimizer for I redν on M + ( I ν ) .In order to prove the monotonicity of γ ( ν ) we consider − a < ν < ν ≤ ν . Let u ∈ M ( I ν ) be a minimizer for γ ( I ν ) and define s > by su + ∈ M + ( I ν ) . Then wehave, with ( I redν , h ν ) denoting the reduction couple for I ν and u := t u + + h ν ( su + ) ∈M ( I ν ) : γ ( I ν ) ≤ I redν ( su + ) = I ν ( u ) = I ν ( u ) − ν − ν | u | ≤ I redν ( t u + ) − ν − ν | u | ≤ max t> I redν ( tu + ) − ν − ν | u | = γ ( I ν ) − ν − ν | u | . Choosing a minimizer v ∈ M ( I ν ) for γ ( I ν ) , defining t > by tv + ∈ M + ( I ν ) , andsetting u := tv + + h ν ( tv + ) ∈ M ( I ν ) , an analogous argument shows that γ ( I ν ) ≤ γ ( I ν ) + ν − ν | u | . For the continuity of γ ( ν ) it remains to prove that s, t are bounded for ν , ν in a compactsubset of ( − a, ν ] because then | γ ( I ν ) − γ ( I ν ) | = O ( ν − ν ) . This follows for s from < I redν ( su + ) ≤ s (cid:0) k u + k + ν | u + | (cid:1) − d p λp s p | u + | pp . where d p > is from Proposition 3.1. The bound for t is proved analogously. For a subset Λ ⊂ R , let Λ c denote its complement, and Λ ε := (cid:8) x ∈ R : εx ∈ Λ (cid:9) , ε > . By the change of variables x εx and setting V ε ( x ) = V ( εx ) , the singularlyperturbed problem (1.1) is equivalent to − iα · ∇ u + aβu + V ε ( x ) u = f ( | u | ) u. (5.1)In the sequel, we will modify the function f similar to [9, 10]. For < δ ≤ a − | V | ∞ , (5.2)we define ˜ f = ˜ f δ ∈ C ( R +0 ) by ˜ f (0) = 0 and dds (cid:0) ˜ f ( s ) s (cid:1) = min (cid:8) f ′ ( s ) s + f ( s ) , δ (cid:9) . In the subcritical case κ = 0 of Theorem 2.1 the choice δ = a −| V | ∞ will be fine. For thecritical case κ > we need to make δ smaller in the course of the proof. Let ˜ F ( s ) = R s ˜ f ( t ) t dt be the primitive of ˜ f ( s ) s . By our assumptions on V there exists R > so that ∇ V ( x ) / ∈ R x for all x ∈ R with | x | = R and V ( x ) = V (0) , (5.3)4see [8]. We define the cut-off function χ : R → [0 , by χ ( x ) = , if | x | < R R −| x | R , if R ≤ | x | < R , if | x | ≥ R . (5.4)and consider g ( x, s ) = χ ( x ) f ( s ) + (cid:0) − χ ( x ) (cid:1) ˜ f ( s ) as well as G ( x, s ) = Z s g ( x, t ) tdt = χ ( x ) F ( s ) + (cid:0) − χ ( x ) (cid:1) ˜ F ( s ) . For later use, associated to the above notations, we denote B = B (0 , R ) and B = B (0 , R ) the open balls in R of radius R and R . The following lemma is easy toprove. Lemma 5.1.
The function G ( x, s ) satisfies the conditions ( i ) − ( iii ) from Theorem . We will consider the truncated problem − iα · ∇ u + aβu + V ε ( x ) u = g ε ( x, | u | ) u, u ∈ E (5.5)where we write g ε ( x, s ) = g ( εx, s ) ; we also use the notations χ ε and G ε for the dilationsof χ and G , respectively. The corresponding energy functional is Φ ε ( u ) = 12 (cid:0) k u + k − k u − k (cid:1) + 12 Z R V ε ( x ) | u | dx − Z R G ε ( x, | u | ) dx. As a direct consequence of Lemma 5.1, we can introduce (Φ redε , h ε ) as the reductioncouple of Φ ε .In order to establish a compactness result for Φ ε , we first prove a bound for Palais-Smale sequences of Φ ε that is uniform in ε . Lemma 5.2.
For c ∈ R fixed, ( P S ) c -sequences of Φ ε are bounded uniformly in ε .Proof. Given a ( P S ) c -sequence ( u n ) n for Φ ε we have by our conditions on f : Z R χ ε ( x ) f ( | u n | ) | u n | · | u + n − u − n | dx ≤ (cid:16) Z R χ ε ( x ) (cid:0) f ( | u n | ) | u n | (cid:1) dx (cid:17) · (cid:12)(cid:12) u + n − u − n (cid:12)(cid:12) + δ Z R χ ε ( x ) | u n | · | u + n − u − n | dx ≤ C θ (cid:16) Z R χ ε ( x ) (cid:0) f ( | u n | ) | u n | − F ( | u n | ) (cid:1) dx (cid:17) k u n k + δ | u n | , C θ > only depends on the constant θ > in ( f . It follows from (5.2) that (cid:16) − | V | ∞ a (cid:17) k u n k ≤ Φ ′ ε ( u n )[ u + n − u − n ] + Z R g ε ( x, | u n | ) | u n | · | u + n − u − n | dx ≤ C θ (cid:16) ε ( u n ) − Φ ′ ε ( u n )[ u n ] (cid:17) k u n k + 2 δ | u n | + o ( k u n k ) . Now the lemma follows using (3.3): (cid:16) − | V | ∞ + 2 δ a (cid:17) k u n k ≤ C θ (cid:0) c + o (1)) + o ( k u n k ) (cid:1) k u n k + o ( k u n k ) . (5.6)Now we can prove the Palais-Smale condition for Φ ε . Recall that the nonlinearity G in Φ ε depends on a constant δ ; see (5.2). Proposition 5.3. If κ · c < (cid:18) a − | V | ∞ a (cid:19) · S , then there exists δ > such that the truncated functional Φ ε satisfies the ( P S ) c -conditionfor all c ≤ c , all ε > .Proof. We choose δ > so that (cid:18) a − | V | ∞ a (cid:19) S > (cid:18) a − ( | V | ∞ + δ ) a (cid:19) S > κ · c . Let ( u n ) n be a ( P S ) c -sequence for Φ ε with c ≤ c , any ε > . By Lemma 5.2 there exists u ∈ E such that, along a subsequence, u n ⇀ u in E and u n → u strongly in L qloc for q ∈ [2 , . We want to show that u n → u strongly in E .Set z n = u n − u so that z n ⇀ in E and k u ± n k = k u ± k + k z ± n k + o n (1) . Note that lim s → ˜ f ( s ) = lim s →∞ ˜ f ( s ) s = 0 and lim s → f ( s ) = lim s →∞ f ( s ) s − κ = 0 . By the Brezis-Lieb lemma (see for instance [45, Lemma 1.32]) there holds Z R G ε ( x, | u n | ) = Z R G ε ( x, | u | ) + Z R (cid:0) − χ ε ( x ) (cid:1) ˜ F ( | z n | ) + κ Z R χ ε ( x ) | z n | + o n (1) , and Z R g ε ( x, | u n | ) | u n | = Z R g ε ( x, | u | ) | u | + Z R (cid:0) − χ ε ( x ) (cid:1) ˜ f ( | z n | ) | z n | + κ Z R χ ε ( x ) | z n | + o n (1) . Therefore Φ ε ( u n ) = Φ ε ( u ) + Φ ε ( z n ) + o n (1) , D Φ ε ( u n )[ u n ] = D Φ ε ( u )[ u ] + D Φ ε ( z n )[ z n ] + o n (1) . Obviously, D Φ ε ( u ) = 0 , hence D Φ ε ( z n )[ z n ] = o n (1) . We claim that D Φ ε ( z n ) → as n → ∞ . (5.7)In fact, consider ϕ ∈ E with k ϕ k ≤ and set g ( x, s ) = g ( x, s ) − κχ ( x ) s . We have D Φ ε ( u n )[ ϕ ] = (cid:10) u + n − u − n , ϕ (cid:11) + Re Z R V ε ( x ) u n · ¯ ϕ − Re Z R g ε ( x, | u n | ) u n · ¯ ϕ = (cid:10) z + n , ϕ + (cid:11) − (cid:10) z − n , ϕ − (cid:11) + (cid:10) u + , ϕ + (cid:11) − (cid:10) u − , ϕ − (cid:11) + Re Z R V ε ( x ) z n · ¯ ϕ + Re Z R V ε ( x ) u · ¯ ϕ − Re Z R g ε ( x, | z n | ) z n · ¯ ϕ − Re Z R g ε ( x, | u | ) u · ¯ ϕ − κ · Re Z R χ ε ( x ) | z n + u | ( z n + u ) · ¯ ϕ + o n ( k ϕ k ) (5.8)where we used u n = z n + u and D Φ ε ( u ) = 0 . The estimate for the subcritical partRe Z R g ε ( x, | u n | ) u n · ¯ ϕ − Re Z R g ε ( x, | z n | ) z n · ¯ ϕ − Re Z R g ε ( x, | u | ) u · ¯ ϕ = o n ( k ϕ k ) follows from a standard argument in [12, Lemma 7.10]. To estimate the last integral in(5.8), we set ψ n := | z n + u | ( z n + u ) − | z n | z n − | u | u and observe | ψ n | ≤ | z n | · | u | . Bythe Egorov theorem there exists Θ σ ⊂ B ε such that meas ( B ε \ Θ σ ) < σ and z n → uniformly on Θ σ as n → ∞ . Thus, by the H¨older inequality, we have Z R χ ε ( x ) | ψ n | · | ϕ | ≤ Z Θ σ | ψ n | · | ϕ | + Z B ε \ Θ σ | ψ n | · | ϕ |≤ Z Θ σ | ψ n | · | ϕ | + 2 Z B ε \ Θ σ | z n | ! · Z B ε \ Θ σ | u | ! · Z B ε \ Θ σ | ϕ | ! . The first integral in the last line converges to as n → ∞ and the remaining integrals goto uniformly in n as σ → . This shows Z R χ ε ( x ) | ψ n | · | ϕ | = o n ( k ϕ k ) as n → ∞ D Φ ε ( u ) = 0 , D Φ ε ( u n )[ ϕ ] = (cid:10) z + n , ϕ + (cid:11) − (cid:10) z − n , ϕ − (cid:11) + (cid:10) u + , ϕ + (cid:11) − (cid:10) u − , ϕ − (cid:11) + Re Z R V ε ( x ) z n · ¯ ϕ + Re Z R V ε ( x ) u · ¯ ϕ − Re Z R g ε ( x, | z n | ) z n · ¯ ϕ − Re Z R g ε ( x, | u | ) u · ¯ ϕ − κ · Re Z R χ ε ( x ) | z n | z n · ¯ ϕ − κ · Re Z R χ ε ( x ) | u | u · ¯ ϕ + o ( k ϕ k )= D Φ ε ( z n )[ ϕ ] + D Φ ε ( u )[ ϕ ] + o n ( k ϕ k )= D Φ ε ( z n )[ ϕ ] + o n ( k ϕ k ) It follows that D Φ ε ( z n ) → as n → ∞ as claimed in (5.7). Now D Φ ε ( z n )[ z + n − z − n ] = o n (1) reads as k z n k + Re Z R V ε ( x ) z n · ( z + n − z − n )= Z R (cid:0) − χ ε ( x ) (cid:1) ˜ f ( | z n | ) z n · ( z + n − z − n ) + κ · Re Z R χ ε ( x ) | z n | z n · ( z + n − z − n ) + o n (1) . Then, by using the fact ˜ f ( s ) ≤ δ and (4.6), we obtain (cid:18) a − ( | V | ∞ + δ ) a (cid:19) S (cid:18)Z R χ ε ( x ) | z n | dx (cid:19) ≤ κ · Z R χ ε ( x ) | z n | dx + o n (1) . If b := lim n →∞ R R χ ε ( x ) | z n | dx = 0 then k z n k = o n (1) and u n → u strongly in E , asclaimed. Suppose to the contrary that b > so that (cid:18) a − ( | V | ∞ + δ ) a (cid:19) S ≤ κ · Z R χ ε ( x ) | z n | dx = κ · b + o n (1) . In case κ = 0 , this is a contradiction. In case κ > , using Φ ε ( u ) = Z R g ε ( x, | u | ) | u | − G ε ( x, | u | ) ≥ , as well as Φ ε ( u n ) ≥ Φ ε ( z n )+ o n (1) and D Φ ε ( z n )[ z n ] = o n (1) , we obtain the contradiction κ · c + o n (1) ≥ κ · b + o n (1) ≥ (cid:18) a − ( | V | ∞ + δ ) a (cid:19) S o n (1) . We finish this section with a couple of notations that will be of use later. For simplicity,when ν belongs to the range of V ( x ) that is ν ∈ { V ( x ) : x ∈ R } , we denote ν = V (0) and correspondingly I ν = I V (0) , I redν = I redV (0) , γ ( I ν ) = γ ( I V (0) ) . (5.9)8Moreover, given arbitrarily y ∈ R , we can define the functional Φ y : E → R , Φ y ( u ) = 12 (cid:0) k u + k − k u − k (cid:1) + V ( y )2 | u | − Z R G ( y, | u | ) dx, and (Φ redy , h y ) the reduction couple associated to Φ y . Plainly, the critical point of Φ y aresolutions of the problem − iα · ∇ u + aβu + V ( y ) u = g ( y, | u | ) u. When y ∈ B , we have Φ y = I V ( y ) and h y = h V ( y ) . Let us point out that, by virtueof [19, Lemma 4.3], we can conclude the following splitting type result, whose proof ispostponed to the appendix. Proposition 5.4.
For y ∈ R , let us define the functional Φ ε,y : E → R , Φ ε,y ( u ) = 12 (cid:0) k u + k − k u − k (cid:1) + 12 Z R V ( εx + y ) | u | dx − Z R G ( εx + y, | u | ) dx, and (Φ redε,y , h ε,y ) the associated reduction couple, we have that (1) let { y ε } ⊂ R be such that y ε → y for some y ∈ R then, up to a subsequence, h ε,y ε ( w ) → h y ( w ) as ε → for each w ∈ E + ; (2) let { y ε } ⊂ R be such that y ε → y for some y ∈ R and let { w ε } ⊂ E + be suchthat w ε ⇀ w for some w ∈ E + then, up to a subsequence, k h ε,y ε ( w ε ) − h ε,y ε ( w ε − w ) − h y ( w ) k = o ε (1) as ε → ; (3) let { y ε } ⊂ R be such that y ε → y for some y ∈ R and let { w ε } ⊂ E + be suchthat w ε ⇀ w for some w ∈ E + then, up to a subsequence, Φ redε,y ε ( w ε ) − Φ redε,y ε ( w ε − w ) − Φ redy ( w ) = o ε (1) and D Φ redε,y ε ( w ε )[ ϕ ] − D Φ redε,y ε ( w ε − w )[ ϕ ] − D Φ y ( w )[ ϕ ] = o ε (1) k ϕ k uniformly for ϕ ∈ E + as ε → . In this section, we will prove the existence of solutions to the truncated problem (5.5) and,by virtue of Lemma 4.2, we will restrict ourselves in the barrier ≤ κ < ¯ κ where ¯ κ is9define in (3.8). We would like to emphasize that such a choice of ¯ κ can be interpreted aswe choose c = γ ( J ~µ V ) in Proposition 5.3. With all these notations, for such choice of κ , we can fix the constant δ > properly small so that the Palais-Smale condition holdsautomatically in the energy range Φ ε ≤ γ ( J ~µ V ) .To begin with, let us mention that, under our hypotheses on V , there always exists avector space X ⊂ R such that: ( a ) V | X has a strict local maximum at ; ( b ) V | X ⊥ has a strict local minimum at .In fact, in case ( V , X = R if is local maximum or X = { } if is local minimum,whereas, in case ( V , X is the space spanned by eigenvectors associated to negativeeigenvalues of D V (0) . Let P X : R → X be the orthogonal projection (in the case X = { } , P X is simply the trivial projection).In the next, solutions of (5.5) will be obtained as critical points of Φ ε , and a key ingre-dient for the construction of a min-max scheme is using the reduction couple (Φ redε , h ε ) .However, due to the lack of information on the exact behavior of the reduction map h ε : E + → E − , it seems hopeless to make a ”path of least energy spikes” by properscaling as was employed in [8, 32].Recalling ν = V (0) , let us focus on functions in the subspace E + . Denoted by B := B (0 , R ) for some R < R . Let us choose a minimizer U ∈ M ( I ν ) for γ ( I ν ) andconsider the path p ε : B ε → M + (Φ ε ) defined as p ε ( ξ )( x ) = t ξ,ε U + ( x − ξ ) , x ∈ R , where M + (Φ ε ) = (cid:8) w ∈ E + \ { } : D Φ redε ( w )[ w ] = 0 (cid:9) and t ξ,ε is the unique t > suchthat t ξ,ε U + ( · − ξ ) ∈ M + (Φ ε ) . We also define a family of deformations on M + (Φ ε )Γ ε ≡ (cid:8) ϕ : M + (Φ ε ) → M + (Φ ε ) homeomorphism : ϕ ( p ε ( ξ )) = p ε ( ξ ) if ξ ∈ ∂B ε ∩ X (cid:9) . Then we define the min-max level γ ε := inf ϕ ∈ Γ ε max ξ ∈ B ε ∩ X Φ redε ( ϕ ( p ε ( ξ ))) . (6.1)We point out here that, in the case X = { } , γ ε = γ (Φ ε ) = inf M + (Φ ε ) Φ redε . A technicalpoint we would like to emphasize, which constitutes a crucial difference with min-maxquantity defined in [8], is the fact that the elements p ε ( ξ ) + h ε ( p ε ( ξ )) do not resemble aleast energy solution of I ν since not much is known about the map h ε : E + → E − .0 Proposition 6.1.
There exist ε , δ > such that for any ε ∈ (0 , ε )Φ redε ( p ε ( · )) (cid:12)(cid:12) ∂B ε ∩ X ≤ γ ( J ν ) − δ. Proof.
To simplify notation, we use subscript ” ξ ” to indicate the coordinate translation ofa function u ∈ E , that is, u ξ ( x ) = u ( x − ξ ) . Then, on a fixed bounded interval t ∈ [0 , T ] with some T large, we have Φ redε ( tW ξ ) ≤ (cid:0) k tW ξ k − k h ε ( tW ξ ) k (cid:1) + 12 Z R V ε ( x ) | tW ξ + h ε ( tW ξ ) | dx − Z B ε F ( | tW ξ + h ε ( tW ξ ) | ) dx. Let us first remark that there exists σ > such that V ( ξ ) ≤ ν − σ for all ξ ∈ ∂B ε ∩ X . Since t ∈ [0 , T ] is bounded and R < R , by (1) in Proposition 5.4, h ε ( tW ξ ) = h ε ( tW ) ξ → J V ( ξ ) ( tW ) uniformly in t as ε → . Thus, we deduce Φ redε ( tW ξ ) ≤ J redν − σ ( tW ) + o ε (1) ∀ ξ ∈ ∂B ε ∩ X. Finally, since J ν − σ < J ν strictly on compact subsets, we have that max t> J redν − σ ( tW ) = max t> J ν − σ ( tW + J ν − σ ( tW )) < max t> J ν ( tW + J ν − σ ( tW )) ≤ max t> J redν ( tW ) = γ ( J ν ) , which completes the proof. Proposition 6.2.
We have that lim sup ε → γ ε ≤ γ ( J ν ) . Proof.
It suffices to show that lim sup ε → max ξ ∈ B ε ∩ X Φ redε ( p ε ( ξ )) ≤ γ ( J ν ) . (6.2)In the following we take a sequence ε = ε n → , but we drop the sub-index n for the sakeof clarity. For every ε , there exists a maximum point ξ ε ∈ B ε ∩ X such that max ξ ε ∈ B ε ∩ X Φ redε ( p ε ( ξ )) = Φ redε ( p ε ( ξ ε )) . And we see that Φ redε ( p ε ( ξ ε )) ≤ (cid:0) k t ε W ξ ε k − k h ε ( t ε W ξ ε ) k (cid:1) + 12 Z R V ε ( x ) | t ε W ξ ε + h ε ( t ε W ξ ε ) | dx − Z B ε F ( | t ε W ξ ε + h ε ( t ε W ξ ε ) | ) dx, t ε = t ξ ε ,ε . Since we have { t ε } is bounded (up to a subsequence), we can assumethat t ε → t and εξ ε → ξ ∈ B ∩ X . Then we can conclude that Φ redε ( p ε ( ξ ε )) ≤ (cid:0) k t W k − k J V ( ξ ) ( t W ) k (cid:1) + V ( ξ )2 Z R | t W + J V ( ξ ) ( t W ) | dx − Z R F ( | t W + J V ( ξ ) ( t W ) | ) dx + o ε (1)= J redV ( ξ ) ( t W ) + o ε (1) . Notice that V ( ξ ) ≤ ν , then J redV ( ξ ) ( t W ) ≤ max t> J redν ( tW ) = γ ( J ν ) , and hence (6.2) holds.In the next, we will show that γ ε is a critical value of Φ ε . Motivated by [8, 11], we aregoing to give an estimate from below on γ ε and show that γ ε ≥ γ ( J ν ) + o ε (1) . And inorder to do so, we need to compare γ ε with another auxiliary minimization value. Firstly,set B = B (0 , R ) the open ball of radius R and ζ : R → R be a cut-off function ζ ( x ) = ( x if | x | < R , R x/ | x | if | x | ≥ R , (6.3)and let Q ε : R → X be defined as Q ε ( x ) = P X ( ζ ( εx )) . Then, let us define the barycentertype functional B ε : E \ { } → R , B ε ( u ) = R R Q ε ( x ) | u | θ dx R R | u | θ dx , for u ∈ E \ { } where θ ∈ (2 , is the constant in ( f . Recall that (Φ redε , h ε ) is the reduction couple for Φ ε and M + (Φ ε ) = (cid:8) w ∈ E + \ { } : D Φ redε ( w )[ w ] = 0 (cid:9) , let us consider the followingsubset of functions in M + (Φ ε ) : g M + (Φ ε ) = (cid:8) w ∈ M + (Φ ε ) : B ε ( w ) = 0 (cid:9) . We also define the corresponding auxiliary minimization b ε ≡ inf w ∈ g M + (Φ ε ) Φ redε ( w ) . (6.4)When X is trivial, i.e. X = { } , we have g M + (Φ ε ) = M + (Φ ε ) and then b ε = γ ε .The next lemma shows that b ε is well-defined in general. Lemma 6.3.
There exists ε , ̺ > such that for ε ∈ (0 , ε ) , γ ε ≥ b ε ≥ ̺. B ε are taken over the whole space R . The reason is twofold: firstly, theorthogonal projections associated to the decomposition E = E + ⊕ E − are of convolutiontype with some tempered distributions ρ ± (see an abstract result in [27] for operators thatcommutes with translations), and thus, making the choice of compact-supported functionsin E ± by simply multiplying smooth cut-off functions would be in our situation hopelesssince the convolution with ρ ± do not commute with the multiplication in general. Sec-ondly, the barycenter of an element w ∈ E + does not exhibit the location of the mass ofthose u ∈ E with u + = w . Therefore, it is not enough if we only consider the barycenterintegrations over a bounded domain as was introduced in [8, 11]. Proof of Lemma . Since b ε ≥ ̺ follows directly from ( f − ( f for some ̺ > , weonly need to prove that γ ε ≥ b ε for all small ε .Motivated by [8], let us take an arbitrary ϕ ∈ Γ ε . We define ψ ε : B ∩ X → X as ψ ε ( ξ ) = B ε (cid:0) ϕ ( p ε ( ξ/ε )) (cid:1) . We point out here that, by the definition of Γ ε , ϕ ( p ε ( ξ/ε )) = 0 for all ξ ∈ B ∩ X , and so ψ ε is well defined.For ξ ∈ ∂B ∩ X , it can be seen from the definition of B ε that ψ ε ( ξ ) = ξ + o ε (1) uniformly in ξ ∈ ∂B ∩ X, as ε → . Therefore we can choose ε small enough (independent of ϕ ) so that, for all ε ∈ (0 , ε ) ,deg ( ψ ε , B ∩ X,
0) = deg ( id, B ∩ X,
0) = 1 . Then we can conclude that for every ε , there exists ξ ε ∈ B ∩ X such that ψ ε ( ξ ε ) = 0 .Therefore, since ξ ε /ε ∈ B ε ∩ X , there follows max ξ ∈ B ε ∩ X Φ redε ( ϕ ( p ε ( ξ ))) ≥ Φ redε ( ϕ ( p ε ( ξ ε /ε ))) ≥ b ε , which concludes the proof. Proposition 6.4.
We have that lim inf ε → b ε ≥ γ ( J ν ) . The proof of this proposition contains the main difficulties of the paper. It will be pre-sented in the next section. Assuming the conclusion for the moment, jointly with Propo-sition 6.2, we can obtain the following3
Proposition 6.5.
We have that lim ε → γ ε = γ ( J ν ) . From Proposition 6.1 and 6.5, we can get γ ε > Φ redε ( p ε ( · )) (cid:12)(cid:12) ∂B ε ∩ X for all small ε > .Recall that we have restricted κ ∈ [0 , ¯ κ ) , it follows that κ · γ ( J ν ) < (cid:16) a −| V | ∞ a (cid:17) S which guarantees the compactness. Thus, by Proposition 5.3, we easily obtain Theorem 6.6.
There exists ε > such that for ε ∈ (0 , ε ) there exists a solution z ε ofthe problem (5.5) . Moreover, Φ redε ( z + ε ) = Φ ε ( z ε ) = γ ε . The proof will be divided into several parts. As a first step, we prove the existence of aminimizer u ε to be auxiliary problem (6.4). Lemma 7.1.
There exists ε > such that for any ε ∈ (0 , ε ) , there exist u ε ∈ E \ { } with B ε ( u + ε ) = 0 and λ ε ∈ X such that − iα · ∇ u ε + aβu ε + V ε ( x ) u ε = g ε ( x, | u ε | ) u ε + (cid:0) λ ε · Q ε ( x ) | u + ε | θ − u + ε (cid:1) + (7.1) and Φ ε ( u ε ) = b ε . Moreover, the sequence { u ε } is bounded in E .Proof. We sketch the proof as follows. For ε > fixed, by the Ekeland variational princi-ple, there exists a sequence { w n } ⊂ g M + (Φ ε ) which is a constrained ( P S ) -sequence for Φ redε at level b ε , moreover, it can be deduced that there exists { λ n } ⊂ X such that Φ redε ( w n ) → b ε , as n → ∞ , (7.2) D Φ redε ( w n ) − ( λ n · Q ε ( x ) | w n | θ − w n ) + | w n | θθ → , as n → ∞ . (7.3)Now, let us set u n = w n + h ε ( w n ) . Since B ε ( u + n ) = B ε ( w n ) = 0 , by (7.2) and (7.3),repeating the arguments of Lemma 5.2, we get that { u n } is bounded in E (uniformly withrespect to ε ) and, therefore, up to a subsequence, it converges weakly to some u ε ∈ E .Since we have assumed ≤ κ < ¯ κ , it follows that b ε ≤ γ ε ≤ γ ( J ν ) + o ε (1) ≤ γ ( J ~µ V ) for small ε . By Proposition 5.3, { u n } converges strongly in E , i.e. u n → u ε as n → ∞ .Note that u ε = 0 , lim inf ε → b ε > , also the sequence λ n is bounded, we have u ε is thedesired minimizer and this concludes the proof. Lemma 7.2.
We have that u + ε χ B ε is non-vanishing. Proof.
We only consider the case κ > since it is much easier when κ = 0 . To thecontrary, we assume that u + ε χ B ε vanishes. Then we have u + ε χ B ε → in L q for all q ∈ (2 , . At this point we first claim that u + ε χ B ε in L . (7.4)Accepting this fact for the moment, let us consider the function t Φ ε ( tu + ε ) and denote t ε > the unique maximum point which realizes its maximum. Then { t ε } isbounded. Set z ε = t ε u + ε ∈ E + , we have that D Φ ε ( z ε )[ z ε ] = 0 and hence k z ε k + Z R V ε ( x ) | z ε | dx = Z R (1 − χ ε ( x )) ˜ f ( | z ε | ) | z ε | dx + κ Z R χ ε ( x ) | z ε | dx + o ε (1) . Since u + ε χ B ε in L , similarly as that was argued in Proposition 5.3, we soon havethat κ Z R χ ε ( x ) | z ε | dx + o ε (1) ≥ (cid:16) a − ( | V | ∞ + δ ) a (cid:17) S . And hence, thanks to our choice of κ ∈ (0 , ¯ κ ) , we get κ Φ ε ( z ε ) = κ (cid:16) Φ ε ( z ε ) −
12 Φ ′ ε ( z ε )[ z ε ] (cid:17) ≥ (cid:16) a − ( | V | ∞ + δ ) a (cid:17) S o ε (1) > κ γ ( J ~µ V ) . Therefore, we have that γ ( J ~µ V ) < Φ ε ( z ε ) ≤ max t> Φ redε ( tu + ε ) = b ε ≤ γ ( J ν ) as ε → which is impossible due to Lemma 4.2.Now, it remains to show (7.4) is valid. Indeed, it follows from Lemma 7.1 that, forsome C > , b ε = Φ ε ( u ε ) = max t> Φ redε ( tu + ε ) ≥ max t> Φ ε ( tu + ε ) ≥ max t> h t (cid:16) − | V | ∞ + δ a (cid:17) k u + ε k − Cκt Z B ε | u + ε | dx i . Then, if u + ε χ B ε → in L as ε → , we can choose T > (independent of ε ) largeenough such that Φ ε ( T u + ε ) > γ ( J ~µ V ) for all small ε > , and we soon have that lim inf ε → b ε ≥ lim inf ε → Φ ε ( T u + ε ) > γ ( J ~µ V ) which is absurd.5 Lemma 7.3.
We have that { λ ε } ⊂ X is bounded.Proof. Let us assume that λ ε = 0 , otherwise we are done. In the sequel, let us set ˜ λ ε = λ ε / | λ ε | . By elliptic regularity arguments we have that u ε ∈ ∩ q ≥ W ,q ( R , C ) and then,jointly with Proposition 3.2, we are allowed to multiply (7.1) by ∂ ˜ λ ε u ε . Then, we haveRe Z R (cid:16) − iα · ∇ u ε + aβu ε + V ε ( x ) u ε − g ε ( x, | u ε | ) u ε (cid:17) · ∂ ˜ λ ε u ε dx = Re Z R λ ε · Q ε ( x ) | u + ε | θ − u + ε · ∂ ˜ λ ε u + ε dx. (7.5)Now, let us evaluate each term of the previous equality. We get Re Z R ∂ ˜ λ ε (cid:2) ( − iα · ∇ u ε ) · u ε (cid:3) dx = 2 Re Z R ( − iα · ∇ u ε ) · ∂ ˜ λ ε u ε dx and so Re Z R ( − iα · ∇ u ε ) · ∂ ˜ λ ε u ε dx = 0 (7.6)Analogously, we have Z R ∂ ˜ λ ε (cid:2) V ε ( x ) | u ε | (cid:3) dx = ε Z R ∂ ˜ λ ε V ( εx ) | u ε | dx + 2 Re Z R V ε ( x ) u ε · ∂ ˜ λ ε u ε dx and so Re Z R V ε ( x ) u ε · ∂ ˜ λ ε u ε dx = − ε Z R ∂ ˜ λ ε V ( εx ) | u ε | dx = O ( ε ) . (7.7)It also follows that Re Z R aβu ε · ∂ ˜ λ ε u ε dx = 0 . (7.8)For the nonlinear part, let us recall the definition of G ε , ∂ ˜ λ ε G ε ( x, | u ε | ) = ε∂ ˜ λ ε χ ( εx ) (cid:0) F ( | u ε | ) − ˜ F ( | u ε | ) (cid:1) + Re g ε ( x, | u ε | ) u ε · ∂ ˜ λ ε u ε , then we have Z R ∂ ˜ λ ε (cid:2) G ε ( x, | u ε | ) (cid:3) dx = ε Z R (cid:0) F ( | u ε | ) − ˜ F ( | u ε | ) (cid:1)(cid:0) ∂ ˜ λ ε χ ( εx ) (cid:1) dx + Re Z R g ε ( x, | u ε | ) u ε · ∂ ˜ λ ε u ε dx and it follows that Re Z R g ε ( x, | u ε | ) u ε · ∂ ˜ λ ε u ε dx = O ( ε ) . (7.9)6Finally Z R ∂ ˜ λ ε (cid:2) λ ε · Q ε ( x ) | u + ε | θ (cid:3) dx = ε | λ ε | Z B ε | u + ε | θ dx + ε | λ ε | Z R \ B ε R ε | x | h − ( λ ε · x ) | λ ε | | x | i | u + ε | θ dx + θ Re Z R λ ε · Q ε ( x ) | u + ε | θ − u + ε · ∂ ˜ λ ε u + ε dx Observe that ≤ ∂ ˜ λ ε λ ε · Q ε ( x ) ≤ ε | λ ε | for all x ∈ R \ B ε ; this is the key point of ourestimates. And henceRe Z R λ ε · Q ε ( x ) | u + ε | θ − u + ε · ∂ ˜ λ ε u + ε dx = − ε | λ ε | θ Z B ε | u + ε | dx − ε | λ ε | θ Z R \ B ε R ε | x | h − ( λ ε · x ) | λ ε | | x | i | u + ε | dx. (7.10)By (7.5)-(7.10) and Lemma 7.2, we conclude the boundedness of λ ε ∈ X .In what follows, we consider a sequence ε k → and assume that λ ε k → ¯ λ ∈ X . Forsimplicity, we still denote ε k by ε . For a small δ > , let us define H ε = (cid:8) x ∈ R : ¯ λ · Q ε ( x ) ≤ δ (cid:9) . The next proposition gives a complete description of u ε as ε → . We recall thenotations B = B (0 , R ) and B = B (0 , R ) . Proposition 7.4.
Passing to a subsequence if necessary, there exist y ε ∈ H ε , y ∈ B and u ∈ E \ { } with − iα · ∇ u + aβu + V ( y ) u = g ( y , | u | ) u , such that ¯ λ · y = 0 and εy ε → y , k u ε − u ( · − y ε ) k → as ε → . Proof.
We divide the proof into different steps:
Step 1. u + ε | H ε in the L -norm and L θ -norm.Let us first show that u + ε in L θ ( H ε ) . Suppose contrarily that Z H ε | u + ε | θ dx → , as ε → . B ε ( u + ε ) = 0 and ¯ λ ∈ X , we have Z R ¯ λ · Q ε ( x ) | u + ε | θ dx = Z H ε ¯ λ · Q ε ( x ) | u + ε | θ dx + Z H cε ¯ λ · Q ε ( x ) | u + ε | θ dx ≥ δ Z H cε | u + ε | θ dx + Z H ε ¯ λ · Q ε ( x ) | u + ε | θ dx. Therefore δ Z H cε | u + ε | θ dx ≤ (cid:12)(cid:12)(cid:12) Z H ε ¯ λ · Q ε ( x ) | u + ε | θ dx (cid:12)(cid:12)(cid:12) ≤ | ¯ λ | R Z H ε | u + ε | θ dx and so Z H cε | u + ε | θ dx → , as ε → . Then we get u + ε → in L θ which is a contradiction with Lemma 7.2. Now, by theboundedness of { u ε } in E and so in L , we can conclude by interpolation: for a suit-able µ ∈ (0 , < c ≤ k u + ε k L θ ( H ε ) ≤ k u + ε k µL ( H ε ) k u + ε k − µL ( H ε ) ≤ C k u + ε k µL ( H ε ) . Step 2.
Passing to be limit by concentration-compactness.By Step 1, we can conclude that { u + ε | H ε } is non-vanishing. And hence, by concentration-compactness arguments (see [34]), there exist y ε ∈ H ε and r > such that Z B ( y ε ,r ) ∩ H ε | u + ε | ≥ c > . Therefore there exits u ∈ E \ { } such that v ε = u ε ( · + y ε ) ⇀ u in E . Claim 7.1. { εy ε } is bounded and, up to a subsequence, εy ε → y ∈ B as ε → . To see this, let us assume that εy ε B and dist( εy ε , ∂B ) /ε → ∞ . Observe that v ε solves the equation − iα · ∇ v ε + aβv ε + V ( εx + εy ε ) v ε = g ( εx + εy ε , | v ε | ) v ε + (cid:0) λ ε · Q ε ( x + y ε ) | v ε | θ − v ε (cid:1) + , and if we assume that V ( εy ε ) → ν as ε → (passing to a subsequence), we have that u is a weak solution of − iα · ∇ u + aβu + ν u = ˜ f ( | u | ) u + (cid:0) ¯ λ · ˜ y | u + | θ − u + (cid:1) + (7.11)where ˜ y ∈ B is given by ˜ y = lim ε → εy ε if εy ε ∈ B , lim ε → R y ε | y ε | if εy ε ∈ B c . y ε ∈ H ε , we have that ¯ λ · ˜ y ≤ δ and, by the definition of ˜ f , we easily get that ¯ λ · ˜ y > (otherwise u +1 should be ). Now we let e Φ : E → R denote the associatedenergy functional for (7.11), that is e Φ ( u ) = 12 (cid:0) k u + k − k u − k (cid:1) + ν | u | − Z R ˜ F ( | u | ) dx − ¯ λ · ˜ y θ Z R | u + | θ dx. Remark that, for any u ∈ E with u + = 0 and arbitrary v ∈ E , there holds that ¯ λ · ˜ y Z R | u + | θ − | v + | dx + ( θ − λ · ˜ y Z R | u + | θ − (cid:16) | u + | + Re u + · v + | u + | (cid:17) dx > . As a consequence of [1, Theorem 5.1] (see also [19, Lemma 4.6]), we have that Theo-rem 3.3 applies to the situation here. So, we can take ( e Φ red , ˜ h ) to be the reduction couplefor e Φ and let ˜ γ stand for the critical level realized by u , we then have ˜ γ = e Φ red ( u +1 ) = max t> e Φ red ( tu +1 ) ≥ max t> e Φ ( tu +1 ) ≥ max t> t (cid:0) k u +1 k − ( | V | ∞ + δ ) | u | (cid:1) − ¯ λ · ˜ y θ t θ Z R | u +1 | θ dx ≥ max t> t (cid:0) k u +1 k − ( | V | ∞ + δ ) | u | (cid:1) − δθ t θ Z R | u +1 | θ dx. Since k u k ≤ k v ε k = k u ε k < ∞ , we can conclude that ˜ γ > γ ( J ν ) provided that δ isfixed small enough. However, by Fatou’s lemma, we get ˜ γ = e Φ ( u ) − D e Φ ( u )[ u ] = Z R
12 ˜ f ( | u | ) | u | − ˜ F ( | u | ) dx + (cid:0) − θ (cid:1) ¯ λ · ˜ y | u +1 | θθ ≤ Z R
12 ˜ f ( | u | ) | u | − ˜ F ( | u | ) dx + O ( δ ) ≤ O ( δ ) + lim inf ε → Z R g ( εx + εy ε , | v ε | ) | v ε | − G ( εx + εy ε , | v ε | ) dx = O ( δ ) + lim inf ε → Φ ε ( u ε ) ≤ γ ( J ν ) which is impossible. This proves the claim.Now by Claim 7.1, passing to the limit, we have u is a weak solution of − iα · ∇ u + aβu + V ( y ) u = g ( y , | u | ) u + (cid:0) ¯ λ · y | u +1 | θ − u +1 (cid:1) + , with εy ε → y ∈ B such that ¯ λ · y ≤ δ and there exits ¯ c > such that k u ε k ≥ k u k ≥ ¯ c > . Let us define z ,ε = u ε − u ( · − y ε ) . We consider two possibilities: either k z +1 ,ε k → ornot. In the first case the proposition should be proved. In the second case, there are twosub-cases: either z +1 ,ε | H ε → in the L θ -norm or not.9 Step 3.
Assume that z +1 ,ε | H ε in the L θ -norm.In this case, we can repeat the previous argument to the sequence { z ,ε } to obtain y ε ∈ H ε such that Z B ( y ε ,r ) ∩ H ε | z +1 ,ε | ≥ c > . Therefore there exists u ∈ E \ { } such that v ε = z ,ε ( · + y ε ) ⇀ u in E . Moreover, | y ε − y ε | → ∞ , εy ε → y ∈ B , ¯ λ · y ≤ δ and − iα · ∇ u + aβu + V ( y ) u = g ( y , | u | ) u + (cid:0) ¯ λ · y | u +2 | θ − u +2 (cid:1) + , and k u k ≥ ¯ c > . Also, it follows from the weak convergence, k u ε k ≥ k u k + k u k . Let us set z ,ε = u ε − u ( · − y ε ) − u ( · − y ε ) . Suppose that k z +2 ,ε k 6→ and z +2 ,ε | H ε in L θ , then we can argue again as above. And it is all clear that there exists l ∈ N suchthat, after repeating the above argument for l times, we can get that z + l,ε | H ε → in the L θ -norm. Step 4. k z + l,ε k → as ε → .To the contrary let us assume that k z + l,ε k 6→ . Since Q ε ( · ) is bounded, it follows froma standard argument thatRe Z R λ ε · Q ε ( x ) | u + ε | θ − u + ε · ϕ + dx = l X j =1 ¯ λ · y j Re Z R | u + j ( · − y jε ) | θ − u + j ( · − y jε ) · ϕ + dx + Re Z R λ ε · Q ε ( x ) | z + l,ε | θ − z + l,ε · ϕ + dx + o ε (1) k ϕ k , uniformly for ϕ ∈ E as ε → and, particularly, Z R λ ε · Q ε ( x ) | u + ε | θ dx = l X j =1 ¯ λ · y j Z R | u + j | θ dx + Z R λ ε · Q ε ( x ) | z + l,ε | θ dx + o ε (1) . (7.12)Since B ε ( u + ε ) = 0 , together with Proposition 5.4, we can deduce from (7.12) that o ε (1) = k z + l,ε + h ε ( z + l,ε ) k + Re Z R V ε ( x ) (cid:0) z + l,ε + h ε ( z + l,ε ) (cid:1) · (cid:0) z + l,ε − h ε ( z + l,ε ) (cid:1) dx − Re Z R g ε (cid:0) x, | z + l,ε + h ε ( z + l,ε ) | (cid:1)(cid:0) z + l,ε + h ε ( z + l,ε ) (cid:1) · (cid:0) z + l,ε − h ε ( z + l,ε ) (cid:1) dx − Z R λ ε · Q ε ( x ) | z + l,ε | θ dx. (7.13)0Therefore, by ( f and Proposition 3.1, we obtain k z + l,ε + h ε ( z + l,ε ) k ≤ C | z + l,ε + h ε ( z + l,ε ) | ≤ C ′ k z + l,ε + h ε ( z + l,ε ) k , for some C, C ′ > which implies there exists c > such that k z + l,ε + h ε ( z + l,ε ) k ≥ c . Inwhat follows, for simplicity of notation, we denote ¯ z l,ε = z + l,ε + h ε ( z + l,ε ) . By (7.13) again,and a similar argument as in the proof of Lemma 5.2, we get that k ¯ z l,ε k ≤ C θ (cid:16) Z R χ ε ( x ) (cid:0) f ( | ¯ z l,ε | ) | ¯ z l,ε | − F ( | ¯ z l,ε | ) (cid:1) dx (cid:17) | ¯ z + l,ε − ¯ z − l,ε | + C Z R λ ε · Q ε ( x ) | ¯ z + l,ε | θ dx + o ε (1) ≤ C ′ θ (cid:16) redε ( z + l,ε ) − D Φ redε ( z + l,ε )[ z + l,ε ] (cid:17) k ¯ z l,ε k + C Z R λ ε · Q ε ( x ) | ¯ z + l,ε | θ dx + o ε (1) for some C, C θ , C ′ θ > . Remark that ¯ z + l,ε = z + l,ε → in L θ ( H ε ) . Then, it follows from k ¯ z l,ε k ≥ c and ( f that there exists constant c ′ > (independent of R ) such that lim inf ε → (cid:16) Φ redε ( z + l,ε ) − D Φ redε ( z + l,ε )[ z + l,ε ] (cid:17) ≥ c ′ . (7.14)Next, let us distinguish two possible situations. • Case 1. ¯ λ · y j ≥ for all j = 1 , . . . , l .Since B ε ( u + ε ) = 0 , we have that Z H ε λ ε · Q ε ( x ) | u + ε | θ dx + Z H cε λ ε · Q ε ( x ) | u + ε | θ dx. By virtue of z + l,ε | H ε → in the L θ -norm and ¯ λ · y j ≥ for all j = 1 , . . . , l , we know that Z H ε λ ε · Q ε ( x ) | u + ε | θ dx → l X j =1 ¯ λ · y j Z R | u + j | θ dx ≥ , whereas λ ε · Q ε ( x ) ≥ δ > in H cε . Thus we have ¯ λ · y j = 0 , for all j = 1 , . . . , l, and so δ Z H cε | u + ε | θ dx ≤ Z H cε λ ε · Q ε ( x ) | u + ε | θ dx → , as ε → . We also deduce from (7.12) that Z H cε λ ε · Q ε ( x ) | z + l,ε | θ dx → , as ε → . Φ redε ( u + ε ) as Φ redε ( u + ε ) = Φ redε ( z + l,ε ) + l X j =1 T redy j ( u + j ) + o ε (1) . Moreover, we have that D Φ redε ( u + ε )[ u + ε ] = D Φ redε ( z + l,ε )[ z + l,ε ] + l X j =1 D T redy j ( u + j )[ u + j ] + o ε (1) . Since ¯ λ · y j = 0 for all j = 1 , . . . , l , we have u + j ’s are critical points of T redy j . And so, weget the estimate lim inf ε → b ε = lim inf ε → Φ redε ( u + ε ) = lim inf ε → (cid:16) Φ redε ( z + l,ε ) − D Φ redε ( z + l,ε )[ z + l,ε ] (cid:17) + l X j =1 T redy j ( u + j ) . Recall that we have denoted ¯ z l,ε = z + l,ε + h ε ( z + l,ε ) , hence, by (7.14), we have lim inf ε → b ε ≥ c ′ + l X j =1 T redy j ( u + j ) . Since, by Lemma 5.1, we have that T redy j ( w ) ≥ J redV ( y j ) ( w ) , ∀ w ∈ E + , for all j =1 , . . . , l , we can infer that T redy j ( u + j ) ≥ γ ( J V ( y j ) ) , j = 1 , . . . , l. And therefore lim inf ε → b ε ≥ l · min j =1 ,...,l γ ( J V ( y j ) ) + c ′ . Remark that y j ∈ B = B (0 , R ) , by shrinking R if necessary, we can conclude fromthe continuity of the map ν γ ( J ν ) that (cid:12)(cid:12) γ ( J V ( y j ) ) − γ ( J ν ) (cid:12)(cid:12) < c ′ for all j = 1 , . . . , l, and then we obtain lim inf ε → b ε ≥ γ ( J ν ) + 12 c ′ > γ ( J ν ) which contradicts to Proposition 6.2 and Lemma 6.3. • Case 2.
There exists { j , . . . , j k } ⊂ { , . . . , l } such that ¯ λ · y j m < for m = 1 , . . . , k .In this case, similar as that in Case 1, we can apply Proposition 5.4 to obtain Φ redε ( u + ε ) = (cid:16) Φ redε ( z + l,ε ) − D Φ redε ( z + l,ε )[ z + l,ε ] (cid:17) + l X j =1 (cid:16) T redy j ( u + j ) − D T redy j ( u + j )[ u + j ] (cid:17) + o ε (1) . G ( x, s ) , we have T redy j ( u + j ) − D T redy j ( u + j )[ u + j ] ≥ for all j =1 , . . . , l . Then, we conclude that Φ redε ( u + ε ) ≥ (cid:16) Φ redε ( z + l,ε ) − D Φ redε ( z + l,ε )[ z + l,ε ] (cid:17) + k X m =1 (cid:16) T redy jm ( u + j m ) − D T redy jm ( u + j m )[ u + j m ] (cid:17) + o ε (1) . (7.15)To evaluate the above inequality, let us denote M + ( T y jm ) = (cid:8) w ∈ E + \ { } : D T redy jm ( w )[ w ] = 0 (cid:9) , for m = 1 , . . . , k , and t m > be the unique point such that t m u + j m ∈M + ( T y jm ) . Observe that ¯ λ · y j m < , by Step 2 and Step 3, we get that D T redy jm ( u + j m )[ u + j m ] − ¯ λ · y j m Z R | u + j m | θ dx = 0 , and hence we have t m < . Observe that, by applying Lemma 3.5, we have T redy jm ( u + j m ) − D T redy jm ( u + j m )[ u + j m ] > T redy jm ( t m u + j m ) − D T redy jm ( t m u + j m )[ t m u + j m ] . Then, it follows from t m u + j m ∈ M + ( T y jm ) that T redy jm ( u + j m ) − D T redy jm ( u + j m )[ u + j m ] > γ ( J V ( y jm ) ) , for all m = 1 , . . . , k. Finally, by (7.14) and (7.15), we obtain the inequality lim inf ε → b ε = lim inf ε → Φ redε ( u + ε ) ≥ k · min m =1 ,...,k γ ( J V ( y jm ) ) + c ′ . And therefore, as in Case 1, we conclude easily a contradiction.
Step 5.
Complete description of u ε as ε → .As was argued in the previous steps, we know that there exists l ∈ N and, for any j = 1 , . . . , l , y jε ∈ H ε , y j ∈ B and u j ∈ E \ { } such that | y jε − y j ′ ε | → ∞ , if j = j ′ ,εy jε → y j , (cid:13)(cid:13)(cid:13) u + ε − l X j =1 u + j ( · − y jε ) (cid:13)(cid:13)(cid:13) → ,D T redy j ( u + j ) − ¯ λ · y j (cid:0) | u + j | θ − u + j (cid:1) + = 0 . Observe that there strictly holds T redy j ( u + j ) − D T redy j ( u + j )[ u + j ] > γ ( J V ( y j ) ) provided that ¯ λ · y j < . Moreover, Lemma 4.2 implies that γ ( J V ( y j ) ) ≥ γ ( J ν ) − σ forany y j ∈ B , where σ > can be taken arbitrary small by appropriately shrinking R .Therefore, by Proposition 6.2 and Lemma 6.3, we conclude that l = 1 and ¯ λ · y = 0 . Andthus we have k u ε − u ( · − y ε ) k → as ε → which complete the proof.3 Corollary 7.5. y ∈ X ⊥ and lim inf ε → b ε ≥ γ ( J ν ) .Proof. Since B ε ( u + ε ) = 0 , by Proposition 7.4, we get Z R Q ε ( x ) | u + ε ( x ) | θ dx = Z R P X ( ζ ( εx + εy ε )) | u + ε ( x + y ε ) | θ dx → P X ( y ) Z R | u +1 | θ dx. Then y ∈ X ⊥ , and we soon conclude lim inf ε → b ε ≥ γ ( J V ( y ) ) ≥ γ ( J ν ) . This finishes the proof of Proposition 6.4.
In this section, let us study the asymptotic behavior of the solution z ε obtained in The-orem 6.6. We will show that z ε is actually a solution of the original problem (5.1), andconsequently, we can complete the proof of Theorem 2.1.Let us recall that z ε is the critical point of Φ ε at level γ ε , that is, − iα · ∇ z ε + aβz ε + V ε ( x ) z ε = g ε ( x, | z ε | ) z ε . (8.1)Moreover, Proposition 6.5 implies that Φ ε ( z ε ) → γ ( J ν ) as ε → .In what follows, we will give the asymptotic behavior of z ε as ε → . Proposition 8.1.
Given a sequence ε j → , up to a subsequence, there exists { y ε j } ⊂ R such that ε j y ε j → , k z ε j − Z ( · − y ε j ) k → , where Z ∈ L ν ( see (5.9) ) .Proof. For the sake of clarity, let us write ε = ε j . Our argument here has been usedalready in the previous section, so we will be sketchy. First of all, analogous to Proposition7.4, we can conclude that: there exist ¯ y ε ∈ R , ¯ y ∈ B and z ∈ E \ { } with − iα · ∇ z + aβz + V (¯ y ) z = g (¯ y , | z | ) z , such that ε ¯ y ε → ¯ y , k z ε − z ( · − y ε ) k → as ε → . So, the only thing that need to be proved is that ¯ y = 0 .4 By regularity arguments, { z ε } ⊂ ∩ q ≥ W ,q ( R , C ) . For arbitrary ξ ∈ R , multiplying(8.1) by ∂ ξ z ε and integrating, we get − ε Z R ∂ ξ V ( εx ) | z ε | dx + ε Z R (cid:0) F ( | z ε | ) − ˜ F ( | z ε | ) (cid:1) ∂ ξ χ ( εx ) dx = 0 . (8.2)And if χ is C around ¯ y , we shall divide by ε and pass to the limit to obtain − ∂ ξ V (¯ y )2 Z R | z | dx + ∂ ξ χ (¯ y ) Z R (cid:0) F ( | z | ) − ˜ F ( | z | ) (cid:1) dx = 0 . (8.3)At this point, similar as that in [8], we consider three different cases. • Case 1. ¯ y ∈ B .By (8.3), we get that ∂ ξ V (¯ y ) = 0 . Since ξ ∈ R is arbitrary, ¯ y is a critical point of V in B , and therefore ¯ y = 0 . • Case 2. ¯ y ∈ B \ B .In this case, let us first fix ξ = | ¯ y | ¯ y . By the definition of χ (see (5.4)), we have that ∂ ξ χ (¯ y ) = − /R .Now, using ( f and the fact ˜ F ( s ) ≤ δ s , it follow easily that there exists a constant c > (which is independent of the choice of δ ) such that Z R F ( | z | ) dx ≥ c, and so by the boundedness of z ∈ E (see an argument of Lemma 5.2) we get c ′ Z R | z | dx ≤ Z R (cid:0) F ( | z | ) − ˜ F ( | z | ) (cid:1) dx. Thus, it suffices to take R smaller, if necessary, to get a contradiction with (8.3). • Case 3. ¯ y ∈ ∂B .In this case, observe that χ (¯ y ) = 1 , and so z is a solution of − iα · ∇ z + aβz + V (¯ y ) z = f ( | z | ) z . Since J redV (¯ y ) ( z +1 ) = J V (¯ y ) ( z ) = γ ( J ν ) , Lemma 4.2 implies that V (¯ y ) = ν . Then, by(5.3), there exists τ ∈ R tangent to ∂B at ¯ y such that ∂ τ V (¯ y ) = 0 .Remark that χ is not C on ∂B , let us go back to consider (8.2). Take ξ = τ and r
A Appendix
Here we sketch the proof of Proposition 5.4. Firstly for later use let us point out that,under the assumptions of Proposition 5.4, V ( ε · + y ε ) → V ( y ) in L ∞ loc ( R N ) as ε → .Now, denote V ε ( x ) = V ( εx + y ε ) − V ( y ) , we soon have Φ ε,y ε ( u ) = T y ( u ) + 12 Z R V ε ( x ) | u | dx − Z R (cid:0) G ( εx + y ε , | u | ) − G ( y, | u | ) (cid:1) dx (A.1)for all u ∈ E . We also remark that, for arbitrary w ∈ E + and v ∈ E − , by setting ˜ v = v − h ε,y ε ( w ) and ℓ ( t ) = Φ ε,y ε (cid:0) w + h ε,y ε ( w )+ t ˜ v (cid:1) , one has ℓ (1) = Φ ε,y ε ( w + v ) , ℓ (0) =Φ ε,y ε (cid:0) w + h ε,y ε ( w ) (cid:1) and ℓ ′ (0) = 0 . Hence we deduce ℓ (1) − ℓ (0) = R (1 − s ) ℓ ′′ ( s ) ds .And consequently, we have Z (1 − s )Ψ ′′ ε,y ε (cid:0) w + h ε,y ε ( w ) + s ˜ v (cid:1) [˜ v, ˜ v ] ds + 12 k ˜ v k + 12 Z R N V ( εx + y ε ) | ˜ v | dx = Φ ε,y ε (cid:0) w + h ε,y ε ( w ) (cid:1) − Φ ε,y ε ( z + v ) , (A.2)6where, for notation convenience, we denote Ψ ε,y ( u ) ≡ R R G ( εx + y, | u | ) dx for u ∈ E and y ∈ R .Observe that assertion (1) follows directly from [19, Lemma 4.3] and that assertion (3) can be viewed as an immediate corollary of assertion (2) . Hence, to complete the proof,it suffices to show that, as ε → , ( y ε → y in R w ε ⇀ w in E + = ⇒ k h ε,y ε ( w ε ) − h ε,y ε ( w ε − w ) − h y ( w ) k = o ε (1) . (A.3)To this end, we first claim that y ε → y in R and u ε ⇀ u in E as ε → ⇒ Φ ε,y ε ( u ε ) − Φ ε,y ε ( u ε − u ) − Φ ε,y ε ( u ) = o ε (1) as ε → . (A.4)This can be proved similarly as (5.8) in Proposition 5.3, therefore we omit the details. Weonly point out here that, for the nonlinear part, it suffices to check Z R (cid:0) G ( εx + y ε , | u ε | ) − G ( εx + y ε , | u ε − u | ) − G ( εx + y ε , | u | ) (cid:1) dx = o ε (1) where G ( x, s ) = G ( x, s ) − κ χ ( x ) s . Since G is subcritical, the proof follows from astandard argument in [12, Lemma 7.10].As a direct consequence of (A.4), we soon conclude thatFor any sequence w ε ⇀ in E + , we have that h ε,y ε ( w ε ) ⇀ in E − . (A.5)Indeed, notice that h ε,y ε ( w ε ) is bounded (see Theorem 3.3), we may assume up to a sub-sequence that h ε,y ε ( w ε ) ⇀ u ∈ E − . Then u ε ≡ w ε + h ε,y ε ( w ε ) ⇀ u . Now, remark that Ψ ε,y ε ≥ , we conclude from (A.4) that a − | V | ∞ a k u k ≤ − Φ ε,y ε ( u ) = Φ ε,y ε ( u ε − u ) − Φ ε,y ε ( u ε ) + o ε (1) ≤ o ε (1) as ε → . And hence u = 0 .Now we are ready to show (A.3). Let w ε ⇀ w in E + . We may assume h ε,y ε ( w ε ) ⇀ v in E − . By (A.5), there holds h ε,y ε ( w ε − w ) ⇀ . Using (A.4) and assertion (1) (i.e. thefact that h ε,y ε ( w ) → h y ( w ) as ε → ), we conclude that Φ ε,y ε (cid:0) w ε + h ε,y ε ( w ε ) (cid:1) = Φ ε,y ε ( w + v ) + Φ ε,y ε (cid:0) w ε − w + h ε,y ε ( w ε ) − v (cid:1) + o ε (1) ≤ Φ ε,y ε (cid:0) w + h ε,y ε ( w ) (cid:1) + Φ ε,y ε (cid:0) w ε − w + h ε,y ε ( w ε − w ) (cid:1) + o ε (1)= Φ ε,y ε (cid:0) w + h y ( w ) (cid:1) + Φ ε,y ε (cid:0) w ε − w + h ε,y ε ( w ε − w ) (cid:1) + o ε (1)= Φ ε,y ε (cid:0) w ε + h ε,y ε ( w ε − w ) + h y ( w ) (cid:1) + o ε (1) as ε → . Now use (A.2), we can deduce that a − | V | ∞ a k h ε,y ε ( w ε ) − h ε,y ε ( w ε − w ) − h y ( w ) k ≤ o ε (1) and hence (A.3) is proved.7 References [1] N. Ackermann, A nonlinear superposition principle and multibump solution of pe-riodic Schr¨odinger equations, J. Funct. Anal. 234 (2006) 423-443.[2] A. Ambrosetti, M. Badiale, S. Cignolani, Semi-classical states of nonlinearShr¨odinger equations, Arch. Rational Mech. Anal. 140 (1997), 285-300.[3] A. 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