Nonlinear Schrödinger equation with unbounded or vanishing potentials: solutions concentrating on lower dimensional spheres
aa r X i v : . [ m a t h . A P ] S e p NONLINEAR SCHR ¨ODINGER EQUATION WITH UNBOUNDEDOR VANISHING POTENTIALS: SOLUTIONS CONCENTRATINGON LOWER DIMENSIONAL SPHERES
DENIS BONHEURE, JONATHAN DI COSMO, AND JEAN VAN SCHAFTINGEN
Abstract.
We study positive bound states for the equation − ε ∆ u + V ( x ) u = K ( x ) f ( u ) , x ∈ R N , where ε > V and K are radial positive potentials.We are especially interested in solutions which concentrate on a k -dimensionalsphere, 1 ≤ k ≤ N −
1, as ε →
0. We adopt a purely variational approachwhich allows us to consider broader classes of potentials than those treatedin previous works. For example, V and K might be singular at the origin orvanish superquadratically at infinity. Introduction
We consider the nonlinear Schr¨odinger equation i ~ ∂ψ∂t = − ~ m ∆ ψ + W ( x ) ψ − | ψ | p − ψ, ( t, x ) ∈ R × R N , (1)which appears for instance in nonlinear optics or condensed matter physics. A standing wave solution of (1) is a solution of the form ψ ( t, x ) = e − iEt/ ~ u ( x ) , where E is the energy of the wave. The function ψ is a standing wave solution of(1) if and only if u is a solution of the semilinear elliptic equation − ε ∆ u + V ( x ) u = | u | p − u, x ∈ R N , (2)where ε = ~ / m and V ( x ) = W ( x ) − E . It is a bound state if u ∈ H ( R N ). Froma physical point of view, one expects to recover the laws of classical mechanics when ~ →
0. It is thus interesting to study the behaviour of the solutions of (2) as ε tends to 0. The bound states of (2) with ε small are referred to as semiclassicalstates .It is well known that problem (2) possesses solutions which exhibit concentra-tion phenomena as ε →
0. More precisely, these solutions converge uniformly to 0outside some concentration set, while remaining uniformly positive in the concen-tration set. This concentration set can be either a point, a finite set of points or amanifold.The solutions concentrating around one or several isolated points have beenintensively studied (see for example [1, 8] and their bibliographies).On the other hand, one can ask if there exist solutions of (2) concentrating ona higher dimensional set. This problem has been solved for some specific higher
Date : February 4, 2010.1991
Mathematics Subject Classification.
Key words and phrases.
Stationary nonlinear Schr¨odinger equation; semiclassical states; semi-linear elliptic problem; singular potential; vanishing potentiel; radial solution; concentration onsubmanifolds.Jonathan Di Cosmo is a research fellow of the Fonds de la Recherche Scientifique–FNRS. dimensional sets. Solutions concentrating on curves have been found recently in [12],see also [11] for the case N = 2 and [13, 14] for a Neumann singularly perturbedproblem. Here we shall restrict ourselves to the problem of solutions concentratingaround spheres. In several recent papers [3–7,9], solutions concentrating on ( N − N − k -dimensional sphere in R N , 1 ≤ k ≤ N −
1. The existence of such solutions has been discussed in remarks in[2], [1], [9]. Particular problems arise in the critical frequency case, namely wheninf R N V = 0. These problems have been tackled in [4] and [9]. Theorem 1 (Ambrosetti-Ruiz [4]) . Assume that p > , that V ∈ C ( R N ) is apositive bounded radially symmetric potential, that ∇ V is bounded and that lim inf | x |→∞ V ( x ) | x | > . If there exists r ∗ such that the function M : (0 , ∞ ) → R defined for r > by M ( r ) := r N − [ V ( r )] p +1 p − − (3) has an isolated local maximum or minimum at r = r ∗ , then, for ε > smallenough, equation (2) has a positive radially symmetric solution u ε ∈ H ( R N ) thatconcentrates at the sphere | x | = r ∗ . The problem in [9] is rather different. The potential V vanishes and the solutionsconcentrate around zeroes of V . The asymptotic behaviour depends on the shapeof V around 0.Theorem 1 relies on a Lyapunov-Schmidt reduction method. The aim of thisnote is to examine possible improvements in the previous results that can be ob-tained by using the penalization method, a variational method originally due toDel Pino and Felmer [10] and adapted to our framework in the papers [8, 16]. Thismethod permits us to treat superquadratically decaying potentials, or even com-pactly supported potentials.Our results include the following simple particular case. Theorem 2.
Let N ≥ , p > NN − and V ∈ C ( R N \ { } , R + ) be a radial potential.If there exists r ∗ > such that the function M ( r ) defined by (3) has an isolated localminimum at r = r ∗ such that M ( r ∗ ) > , then for ε small enough, the equation (2) has a positive radially symmetric solution u ε that concentrates on the sphere ofradius r ∗ . If N ≥
5, one has also that u ε ∈ L ( R N ) (see Corollary 5.7).In contrast with Theorem 1, we do not require any boundedness assumption on V or its derivatives, and we treat potentials V which are singular at the origin orvanish superquadratically at infinity.Theorem 2 is a particular case of Theorem 3 below, which deals with a nonlin-earity which is neither necessarily homogeneous nor autonomous, see equation (4)below. Furthermore, we will find solutions concentrating on a k -dimensional sphere,1 ≤ k ≤ N −
1. In this case, the critical exponent to be taken into considerationis p k = N − k +2 N − k − if N − k ≥ p k = ∞ if N − k = 1 ,
2. We also obtain results for N = 2 with a little more care, see Section 6.Let us point out that if V is compactly supported and p ≤ NN − , then equation(2) has no positive solution in the neighborhood of infinity, see the discussion in[16].Assuming that the potential V is cylindrically symmetric, we can reduce (2)to a problem in R N − k . The single-peaked solutions of this problem can then be ONLINEAR SCHR ¨ODINGER EQUATION 3 extended to R N by symmetry. In this way, we obtain a solution of (2) concentratingaround a k -dimensional sphere. Observe that since the reduced problem is in R N − k ,the critical exponent to be considered is the one in dimension N − k . This allowsfor example to treat critical problems by looking for cylindrically symmetric (nonnecessarily radial) solutions.2. Assumptions and main result
We shall study the equation with a more general nonlinearity − ε ∆ u + V ( x ) u = K ( x ) f ( u ) , x ∈ R N . (4) Let k be a fixed integer such that 1 ≤ k ≤ N −
1. This number k is the dimensionof the sphere on which we want to construct concentrating solutions. Let us chooseany ( N − k − H ⊂ R N . We denote by H ⊥ theorthogonal complement of H . If x ∈ R N , we will write x = ( x ′ , x ′′ ) with x ′ ∈ H and x ′′ ∈ H ⊥ .2.1. The potentials.
We consider a nonnegative potential V ∈ C ( R N \ { } ) anda nonnegative competing function K ∈ C ( R N \ { } ), K
0. We assume that forevery R ∈ O ( N ) such that R ( H ) = H , we have V ◦ R = V and K ◦ R = K . Thiswill be the case if for example V and K are radial functions.2.2. The nonlinearity.
We make classical assumptions on f that lead to a goodminimax characterization of the infimum on the Nehari manifold. Namely, weassume that f : R + → R + is continuous and that( f ) there exists q > f ( s ) = O ( s q ) as s → + ,( f ) there exists p > p +1 > − N − k and f ( s ) = O ( s p ) as s → ∞ ,( f ) there exists 2 < θ ≤ p + 1 such that0 < θF ( s ) ≤ f ( s ) s for s > , where F ( s ) := R s f ( σ ) dσ ,( f ) the function s f ( s ) s is nondecreasing.Notice that ( f ) is nothing but the subcriticality condition in dimension N − k .2.3. The growth conditions.
Following [8, 16] we impose one of the three sets ofgrowth conditions at infinity :( G ∞ ) there exists σ < ( N − q − N such thatlim sup | x |→∞ K ( x ) | x | σ < ∞ ;( G ∞ ) there exists σ ∈ R such thatlim inf | x |→∞ V ( x ) | x | > | x |→∞ K ( x ) | x | σ < ∞ ;( G ∞ ) there exist α < σ ∈ R such thatlim inf | x |→∞ V ( x ) | x | α > | x |→∞ K ( x )exp( σ | x | − α ) < ∞ . Note that in comparison with [8], in ( G ∞ ) and ( G ∞ ), V might vanish somewhere.We also impose one of the three sets of growth conditions at the origin, whichmirror those at infinity : DENIS BONHEURE, JONATHAN DI COSMO, AND JEAN VAN SCHAFTINGEN ( G ) there exists τ > −
2, such thatlim sup | x |→ K ( x ) | x | τ < ∞ , ( G ) there exists τ ∈ R such thatlim inf | x |→ V ( x ) | x | > | x |→ K ( x ) | x | τ < ∞ ;( G ) there exist γ > τ ∈ R such thatlim inf | x |→ V ( x ) | x | γ > | x |→ K ( x )exp( τ | x | − γ − ) < ∞ . By Kelvin transform, there is a duality between the conditions at the origin andthe conditions at infinity, at least in the case where f ( t ) = t p . If one defines ˆ u tobe the Kelvin transform of u , i.e.,ˆ u ( x ) = 1 | x | N − u (cid:16) x | x | (cid:17) and the transformed potentialsˆ V ( x ) = 1 | x | V (cid:16) x | x | (cid:17) and ˆ K ( x ) = 1 | x | N +2 − p ( N − K (cid:16) x | x | (cid:17) , the function u ε solves (4) if and only if ˆ u ε solves the same problem with ˆ V and ˆ K in place of V and K . One sees that V , K satisfy ( G i ) if and only if ˆ V and ˆ K satisfy( G i ∞ ).The problem at the origin is in a sense in duality with the one at infinity.Whereas a slow decay of V at infinity does allow a lot of freedom for K , a strongsingularity at the origin allows for very singular K ’s too. The critical thresholdgrowth is 1 / | x | both at the origin and at infinity. This can be made clearer ifwe observe that the optimal barrier functions at the origin are the optimal one atinfinity mapped by Kelvin transform.2.4. The auxiliary potential.
Before we can state our last assumption, we needa few preliminaries. Let a, b >
0. The equation − ∆ u + au = bf ( u ) in R N − k (5)is called the limit equation associated with (4). The weak solutions of (5) arecritical points of the functional I a,b : H ( R N − k ) → R defined by I a,b ( u ) := 12 Z R N − k (cid:16) |∇ u | + au (cid:17) dx − b Z R N − k F ( u ) dx. (6)Any nontrivial critical point u ∈ H ( R N − k ) of I a,b , belongs to the Nehari manifold N a,b := (cid:8) u ∈ H ( R N − k ) | u hI ′ a,b ( u ) , u i = 0 (cid:9) . A solution u ∈ H ( R N − k ) is a least-energy solution of (5) if I a,b ( u ) = inf v ∈N a,b I a,b ( v ) . The ground-energy function is defined by E : R + × R + → R + : ( a, b )
7→ E ( a, b ) := inf u ∈N a,b I a,b ( u ) , ONLINEAR SCHR ¨ODINGER EQUATION 5 and the auxiliary potential M : R N → (0 , + ∞ ] by( x ′ , x ′′ )
7→ M ( x ′ , x ′′ ) := (cid:26) | x ′′ | k E ( V ( x ) , K ( x )) if K ( x ) > , + ∞ if K ( x ) = 0 . The following lemma states some properties of the ground-energy function, see[8, Lemma 3].
Lemma 2.1.
Assume f : R + → R + is a continuous function that fulfills assump-tions ( f )-( f ). Then, for every ( a, b ) ∈ R +0 × R +0 , E ( a, b ) is a critical value of I a,b and we have E ( a, b ) = inf u ∈ H ( R N ) u =0 max t ≥ I a,b ( tu ) . If u ∈ N a,b and E ( a, b ) = I a,b ( u ) , then u ∈ C ( R N ) and up to a translation, u isa radial function such that ∇ u ( x ) · x < for every x ∈ R N \ { } . Moreover, thefollowing properties hold: (i) E is continuous in R +0 × R +0 ; (ii) for every b ∗ ∈ R +0 , a → E ( a, b ∗ ) is strictly increasing; (iii) for every a ∗ ∈ R +0 , b → E ( a ∗ , b ) is strictly decreasing; (iv) for every λ > , E ( λa, λb ) = λ − N/ E ( a, b ) ; (v) if f ( u ) = u p with − N − k < p +1 < , then E ( a, b ) = E (1 , a p +1 p − − N b − p − . If f ( u ) = u p with − N − k < p +1 < , the last property of the preceding lemmaimplies the following explicit form of the auxiliary potential: M ( x ′ , x ′′ ) = E (1 , | x ′′ | k [ V ( x )] p +1 p − − N − k [ K ( x )] − p − . Due to the symmetry that we shall impose on the solution (see (14)), the con-centration can only occur in the space H ⊥ . We assume that there exists a smoothbounded open set Λ ⊂ R N such that(7) ¯Λ ∩ H = ∅ , Λ ∩ H ⊥ = ∅ , for every R ∈ O ( N ) such that R ( H ) = H , R (Λ) = Λ(8)and 0 < inf Λ ∩H ⊥ M < inf ∂ Λ ∩H ⊥ M . (9)In the case where k = N −
2, we shall need the conditioninf Λ ∩H ⊥ M < Λ M . (10)By continuity of M in Λ, this condition is not restrictive. Similarly, we can alsoassume that V > M is continuous on Λ.Our main result is the following theorem. Theorem 3.
Let N ≥ , V, K ∈ C ( R N \ { } , R + ) satisfy one set ( G i ) of growthconditions at the origin and one set ( G j ∞ ) of growth conditions at infinity, and f satisfy assumptions ( f ) - ( f ) . Assume there exists an open bounded set Λ ⊂ R N such that (7) , (8) , (9) and, if k = N − , (10) hold. Then there exists ε > such that for every < ε < ε , problem (2) has at least one positive solution u ε . DENIS BONHEURE, JONATHAN DI COSMO, AND JEAN VAN SCHAFTINGEN
Moreover, for every < ε < ε , there exists x ε ∈ Λ ∩ H ⊥ such that u ε attains itsmaximum at x ε , lim inf ε → u ε ( x ε ) > , lim ε → M ( x ε ) = inf Λ ∩H ⊥ M , and there exist C > and λ > such that u ε ( x ) ≤ C exp (cid:18) − λε d ( x, S kε )1 + d ( x, S kε ) (cid:19) (cid:16) | x | (cid:17) − ( N − , ∀ x ∈ R N , where S kε is the k -sphere centered at the origin and of radius | x ′′ ε | . In the special case where x ∈ Λ ∩ H ⊥ is the unique minimizer of M on Λ ∩ H ⊥ ,then x ε → x , and the solution concentrates around a k –dimensional sphere ofradius | x | centered at the origin.One should note that the theorem is valid in dimension 2, but the solutions thatare obtained do not decay at infinity in general.Theorem 2 follows from Theorem 3 by taking K ≡ f ( u ) = u p and k = N − G ) is always satisfied whereas thecondition ( G ∞ ) holds if and only if ( N − p − N >
0, i.e. p > NN − .The sequel of the paper is devoted to the proof of Theorem 3. In Section 3,we introduce a penalized problem and prove that it has a least energy solution. InSection 4, we study the asymptotics of this solution and in Section 5, we obtaindecay estimates of the solution and show that it also solves the original problem.In all these sections, we assume that N ≥
3. The modifications for the case N = 2will be addressed in Section 6.3. The penalization scheme
We assume that N ≥
3. Let D ( R N ) be the set of compactly supported smoothfunctions. The homogeneous Sobolev space D , ( R N ) is the closure of the set ofcompactly supported smooth functions D ( R N ) with respect to the norm (cid:18)Z R N |∇ u | dx (cid:19) . Thanks to Sobolev inequality, we have D , ( R N ) ⊂ L ∗ ( R N ). Let us also recallHardy’s inequality in R N . One has (cid:18) N − (cid:19) Z R N | u ( x ) | | x | dx ≤ Z R N |∇ u | , for all u ∈ D , ( R N ).Following [16], we define the penalization potential H : R N → R by H ( x ) := κ | x | (cid:16)(cid:0) log | x | (cid:1) + 1 (cid:17) β where β > < κ < ( N − ) . Notice that for all x ∈ R N , we have H ( x ) ≤ κ | x | . By Hardy’s inequality, we deduce that the quadratic form associated to − ∆ − H ispositive, i.e. Z R N (cid:16) |∇ u | − Hu (cid:17) ≥ (cid:18)(cid:16) N − (cid:17) − κ (cid:19) Z R N | u ( x ) | | x | dx ≥ , (11) ONLINEAR SCHR ¨ODINGER EQUATION 7 for all u ∈ D , ( R N ).This inequality implies the following comparison principle. Proposition 3.1.
Let Ω ⊂ R N \{ } be a smooth domain. Let v, w ∈ H (Ω) ∩ C (Ω) be such that ∇ ( w − v ) − ∈ L (Ω) , ( w − v ) − / | x | ∈ L (Ω) and − ∆ w − Hw ≥ − ∆ v − Hv, ∀ x ∈ Ω . (12) If ∂ Ω = ∅ , assume also that w ≥ v on ∂ Ω . Then w ≥ v in Ω .Proof. It suffices to multiply the inequality (12) by ( w − v ) − , integrate by partsand use (11). (cid:3) Fix µ ∈ (0 , g ε : R N × R + → R by g ε ( x, s ) := χ Λ ( x ) K ( x ) f ( s ) + (1 − χ Λ ( x )) min (cid:8)(cid:0) ε H ( x ) + µV ( x ) (cid:1) s, K ( x ) f ( s ) (cid:9) . Let G ε ( x, s ) := R s g ε ( x, σ ) dσ . One can check that g ε is a Carath´eodory functionwith the following properties :( g ) g ε ( x, s ) = o ( s ) , s → + , uniformly in compact subsets of R N .( g ) there exists p > p +1 > − N − k andlim s →∞ g ε ( x, s ) s p = 0 , ( g ) there exists 2 < θ ≤ p + 1 such that0 < θG ε ( x, s ) ≤ g ε ( x, s ) s ∀ x ∈ Λ , ∀ s > , < G ε ( x, s ) ≤ g ε ( x, s ) s ≤ (cid:0) ε H ( x ) + µV ( x ) (cid:1) s ∀ x / ∈ Λ , ∀ s > , ( g ) the function s g ε ( x, s ) s is nondecreasing for all x ∈ R N .We look for a positive solution of the penalized equation − ε ∆ u + V ( x ) u = g ε ( x, u ) in R N ( P ε )in the Hilbert space H V ( R N ) := (cid:26) u ∈ D , ( R N ) | Z R N V u < ∞ (cid:27) endowed with the norm(13) k u k ε := Z R N (cid:16) ε |∇ u | + V u (cid:17) . We will search for a solution of ( P ε ) in the closed subspace(14) H V, H ( R N ) := (cid:8) u ∈ H V ( R N ) | ∀ R ∈ O ( N ) s.t. R ( H ) = H , u ◦ R = u (cid:9) . Define J ε : H V, H ( R N ) → R by J ε ( u ) := 12 Z R N (cid:16) ε |∇ u ( x ) | + V ( x ) | u ( x ) | (cid:17) dx − Z R N G ε ( x, u ( x )) dx. The functional J ε is well defined and of class C ( H V, H ( R N ) , R ). By the principle ofsymmetric criticality [17], critical points are weak solutions of ( P ε ). Furthermore, J ε has the mountain pass geometry. It remains to show that J ε satisfies the Palais-Smale condition. The proof below is inspired from [8]. Recall that a sequence( u n ) n ⊂ H V, H ( R N ) is a Palais-Smale sequence for J ε if J ε ( u n ) ≤ C and J ′ ε ( u n ) → , n → ∞ . DENIS BONHEURE, JONATHAN DI COSMO, AND JEAN VAN SCHAFTINGEN
Proposition 3.2.
For ε sufficiently small, every Palais-Smale sequence for J ε contains a convergent subsequence.Proof. Let ( u n ) n ⊂ H V, H ( R N ) be a Palais-Smale sequence for J ε . It is standardto check, using ( g ), that for ε sufficiently small, the sequence ( u n ) n is bounded in H V, H ( R N ). We infer that, up to a subsequence, u n ⇀ u in H V, H ( R N ).For λ ∈ R + , set A λ := B (0 , e λ ) \ B (0 , e − λ ). Note that H ( x ) ≤ κ | x | | log | x || β . By Hardy’s inequality, we have for λ ≥ Z R N \ A λ Hu n ≤ κλ β Z R N | u n ( x ) | | x | dx ≤ κλ β (cid:18) N − (cid:19) Z R N |∇ u n | . Since ( u n ) n is bounded in H V, H ( R N ), for every δ >
0, there exists ¯ λ ≥ n →∞ Z R N \ A ¯ λ Hu n < δ. (15)Now we claim that for all δ >
0, there exists ˜ λ > n →∞ Z R N \ A ˜ λ V u n < δ. (16)We only sketch the proof, since the arguments are similar to those in [8, Lemma6]. Since ¯Λ ⊂ R N \ { } is compact, there exists λ ≥ ⊂ A λ . Let ζ ∈ C ∞ ( R ) be such that 0 ≤ ζ ≤ ζ ( s ) = (cid:26) | s | ≤ , | s | ≥ . Define a cut-off function η λ ∈ C ∞ ( R N , R ) by η λ ( x ) := ζ (cid:18) log | x | λ (cid:19) . Since h J ′ ε ( u n ) , η λ u n i = o (1) as n → ∞ , we deduce that(17) Z R N (cid:16) ε |∇ u n | + V u n (cid:17) η λ = Z R N g ε ( x, u n ( x )) u n ( x ) η λ ( x ) dx − ε Z R N u n ∇ u n · ∇ η λ + o (1) , as n → ∞ . If λ ≥ λ , η λ = 0 on Λ and it follows from ( g ) that Z R N g ε ( x, u n ( x )) u n ( x ) η λ ( x ) dx ≤ Z R N (cid:0) ε H + µV (cid:1) u n η λ . (18)On the other hand, using Hardy’s inequality, we can show as in [8] that (cid:12)(cid:12)(cid:12)(cid:12)Z R N u n ∇ u n · ∇ η λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cλ k u n k ε . (19)Combining (17), (18) and (19), we get, for every λ ≥ λ , Z R N \ A λ (cid:16) ε |∇ u n | + (1 − µ ) V u n (cid:17) η λ ≤ Z R N (cid:16) ε |∇ u n | + (1 − µ ) V u n (cid:17) η λ ≤ Z R N ε Hu n η λ + Cλ k u n k ε + o (1) . ONLINEAR SCHR ¨ODINGER EQUATION 9
By (15), for λ large enough,lim sup n →∞ Z R N Hu n η λ ≤ lim sup n →∞ Z R N \ A ¯ λ Hu n < δ Conclusion.
We can write(20) k u n − u k ε = J ′ ε ( u n )( u n − u ) − J ′ ε ( u )( u n − u )+ Z R N ( g ε ( x, u n ( x )) − g ε ( x, u ( x ))) ( u n ( x ) − u ( x )) dx. We notice that the first two terms in the right-hand side tend to 0 as n → ∞ . Fix δ > λ > A λ \ Λ and R N \ A λ , where λ = max { ˜ λ, ¯ λ } .By ( g ), one has | g ε ( x, u n ( x )) | ≤ C | u n ( x ) | p . By Rellich Theorem, the embed-ding H V, H (Λ) ֒ → L q (Λ) is compact for all q > q > − N − k . Wecan thus assume that u n → u in L p +1 (Λ). We deduce that g ε ( x, u n ) → g ε ( x, u ) in L q (Λ) as n → ∞ , where q := p +1 p . We conclude from H¨older inequality that Z Λ ( g ε ( x, u n ( x )) − g ε ( x, u ( x ))) ( u n ( x ) − u ( x )) dx → , as n → ∞ . By ( g ), one has | g ε ( x, u n ( x )) | ≤ (cid:0) ε H ( x ) + µV ( x ) (cid:1) | u n ( x ) | for x ∈ A λ \ Λ. ByRellich Theorem, we can assume that u n → u in L ( A λ \ Λ). We deduce that g ε ( x, u n ) → g ε ( x, u ) in L ( A λ \ Λ) as n → ∞ . We conclude as above that Z A λ \ Λ ( g ε ( x, u n ( x )) − g ε ( x, u ( x ))) ( u n ( x ) − u ( x )) dx → , as n → ∞ . Finally, using ( g ), (15) and (16), we obtainlim sup n →∞ Z R N \ A λ | g ε ( x, u n ( x )) − g ε ( x, u ( x )) | | u n ( x ) − u ( x ) | dx ≤ n →∞ Z R N \ A λ ( | g ε ( x, u n ( x )) u n ( x ) | + | g ε ( x, u ( x )) u ( x ) | ) dx ≤ n →∞ Z R N \ A λ (cid:0) ε H + µV (cid:1) (cid:0) u n + u (cid:1) ≤ µ ) δ, since λ ≥ ¯ λ and λ ≥ ˜ λ .Since δ > n →∞ k u n − u k ε = 0 , which ends the proof. (cid:3) We can now state an existence theorem for the penalized problem ( P ε ). Theproof follows from standard arguments. Theorem 4.
Let g : R × R + → R be a Carath´eodory function satisfying ( g ) − ( g ) and V ∈ C ( R N \ { } ) be a nonnegative function. Then, for all ε > , the functional J ε possesses a nontrivial critical point u ε ∈ H V, H ( R N ) , which is characterized by c ε := J ε ( u ε ) = inf u ∈ H V, H ( R N ) \{ } max t> J ε ( tu ) . (21) The function u ε found in Theorem 4 is called a least energy solution of ( P ε ). Bystandard regularity theory, if u ∈ H ( R N ) is a solution of ( P ε ), then u ∈ W ,q loc ( R N )for every q ∈ (1 , ∞ ). In particular, u ∈ C ,α loc ( R N ) for every α ∈ (0 , g ε is not continuous, we cannot achieve a better regularity. Notice also that, by thestrong maximum principle, any nontrivial nonnegative solution u ∈ C ,α loc ( R N ) of( P ε ) is positive in R N . 4. Asymptotics of solutions
In this section we study the asymptotic behaviour as ε → § R N − k + := R N − k − × R + . Proposition 4.1 (Upper estimate of the critical value) . Suppose that the assump-tions of Theorem 4 are satisfied. For ε small enough, the critical value c ε definedin (21) satisfies c ε ≤ ε N − k (cid:0) ω k inf Λ ∩H ⊥ M + o (1) (cid:1) as ε → , where ω k is the volume of the unit sphere in R k +1 . Moreover, the solution u ε of ( P ε ) found in Theorem 4 satisfies, for some C > , k u ε k ε ≤ Cε N − k . Proof.
Let x = (0 , x ′′ ) ∈ Λ ∩ H ⊥ be such that M ( x ) = inf Λ ∩H ⊥ M . Denote by I the functional defined by (6) with a = V ( x ) and b = K ( x ) and w a ground stateof (5). Take η ∈ D (cid:0) R N − k + (cid:1) to be a cut-off function such that 0 ≤ η ≤ η = 1 in aneighbourhood of (0 , | x ′′ | ) and k∇ η k ∞ ≤ C . Consider the test function u ( x ) := η ( x ′ , | x ′′ | ) w (cid:18) x ′ ε , | x ′′ | − | x ′′ | ε (cid:19) . Setting u ( x ′ , x ′′ ) =: v (cid:18) x ′ ε , | x ′′ | − | x ′′ | ε (cid:19) , we compute by a change of variable J ε ( tu )= ω k t Z R N − k − Z ∞− | x ′′ | ε (cid:16) |∇ v | + V ( εy ′ , ερ + | x ′′ | ) v (cid:17) ( ερ + | x ′′ | ) k εdρ ε N − k − dy ′ − ω k Z R N − k − Z ∞− | x ′′ | ε G ( εy ′ , ερ + | x ′′ | , tv )( ερ + | x ′′ | ) k ε dρ ε N − k − dy ′ . For ε small enough, we obtain ε − ( N − k ) J ε ( tu ) ≤ ω k | x ′′ | k I ( tw ) + o (1) . (22)We deduce from (21) that ε − ( N − k ) c ε ≤ max t> ε − ( N − k ) J ε ( tu ) ≤ ω k | x ′′ | k max t> I ( tw ) + o (1)= ω k M ( x ) + o (1) , which is the desired conclusion. (cid:3) ONLINEAR SCHR ¨ODINGER EQUATION 11
Proposition 4.2 (No uniform convergence to 0 in Λ) . Suppose that the assumptionsof Theorem 4 are satisfied and let ( u ε ) ε ⊂ H V, H ( R N ) be positive solutions of ( P ε ) obtained in Theorem 4. Then there exists δ > such that k u ε k L ∞ (Λ) ≥ δ. Proof.
Suppose by contradiction that k u ε k L ∞ (Λ) → ε →
0. Then, ( f ) impliesthat, for all ε sufficiently small, Kf ( u ε ) ≤ µV u ε in Λ. By ( g ), we deduce that − ε (∆ u ε + Hu ε ) + (1 − µ ) V u ε ≤ R N . Proposition 3.1 then implies that u ε ≡ ε sufficiently small, which is impos-sible. (cid:3) By the symmetry imposed on the solution u ε , one can write u ε ( x ′ , x ′′ ) = ˜ u ε ( x ′ , | x ′′ | )with ˜ u ε : R N − k + → R . Since the H V -norm of u ε is of the order ε ( N − k ) / , it is naturalto rescale ˜ u ε ( x ′ , | x ′′ | ) as ˜ u ε ( x ′ ε + εy ′ , | x ′′ ε | + ε | y ′′ | ) around a well-chosen family ofpoints x ε = ( x ′ ε , x ′′ ε ) ∈ R N .The next lemma shows that the sequences of rescaled solutions converge, up toa subsequence, in C ( R N − k ) to a function v ∈ H ( R N − k ). Lemma 4.3.
Suppose that the assumptions of Theorem 4 are satisfied. Let u ε ∈ H V, H ( R N ) be positive solutions of ( P ε ) found in Theorem 4, ( ε n ) n ⊂ R + and ( x n ) n ⊂ R N be sequences such that ε n → and x n = ( x ′ n , x ′′ n ) → ¯ x = (¯ x ′ , ¯ x ′′ ) ∈ ¯Λ as n → ∞ . Set Ω n := R N − k − × i − | x ′′ n | ε n , + ∞ h and let v n : Ω n → R be defined by v n ( y, z ) := ˜ u ε n ( x ′ n + ε n y, | x ′′ n | + ε n z ) , (23) where ˜ u ε n : R N − k + → R is such that u ε n ( x ′ , x ′′ ) = ˜ u ε n ( x ′ , | x ′′ | ) . Then, there exists v ∈ H ( R N − k ) such that, along a subsequence that we still denote by ( v n ) n , v n C ( R N − k ) −→ v. Proof.
First observe that each v n solves the equation − ∆ v n − ε n kz ∂v n ∂z + V ( y n + ε n y, | x ′ n | + ε n z ) v n = g ε n ( y n + ε n y, | x ′ n | + ε n z, v n ) , (24)in Ω n . We infer from Proposition 4.1 that for all n ∈ N , Z Ω n (cid:16) |∇ v n ( y, z ) | + V ( y n + ε n y, | x ′ n | + ε n z ) | v n ( y, z ) | (cid:17) dy dz ≤ C, with C > n .Define a cut-off function η R ∈ D ( R N − k ) such that 0 ≤ η R ≤ η R ( x ) = 1 if | x | ≤ R/ η R ( x ) = 0 if | x | ≥ R and k∇ η R k ∞ ≤ C/R for some
C >
0. Choose( R n ) n such that R n → ∞ and ε n R n →
0. Since ¯ x ∈ Λ and ¯Λ ∩ H = ∅ , one has ε n R n ≤ | x ′′ n | if n is large enough. Define w n ∈ H ( R N − k ) by w n ( y ) := η R n ( y ) v n ( y ) . On the one hand, we notice that Z R N − k w n ≤ Z B (0 ,R n ) v n ≤ B ( x n ,ε n R n ) V Z Ω n V ( y n + ε n y, | x ′ n | + ε n z ) | v n ( y, z ) | dydz. Since V is positive on ¯Λ and continuous on R N , the convergence of x n to a pointin ¯Λ implies that Z R N − k w n ≤ C. (25)On the other hand, we compute in the same way as in [8, Lemma 13] Z R N − k |∇ w n | ≤ C k v n k H ( B (0 ,R n )) . (26)Since k v n k H ( B (0 ,R n )) ≤ C k u ε n k ε , we deduce from (25) and (26) that ( w n ) n is bounded in H ( R N − k ). Since w n solvesequation (24) on B (0 , R n ) for all n , classical regularity estimates yield that for every R > q >
1, sup n ∈ N k v n k W ,q ( B (0 ,R )) < ∞ . (27)Up to a subsequence, we can now assume that ( w n ) n converges weakly in H ( R N − k ) to some function v ∈ H ( R N − k ). By (27), for every compact K ⊂ R N − k , w n converges to v in C ( K ). Moreover, for n large enough, w n = v n in K so that v n → v in C ( K ). (cid:3) In the next two lemmas, we will estimate from below the action of u ε inside andoutside neighbourhoods of points. Since we expect the concentration set to be a k -sphere in R N , the following distance will be useful. For x, y ∈ R N , let d H ( x, y ) := q | x ′ − y ′ | + ( | x ′′ | − | y ′′ | ) . Thus d H ( x, y ) represents the distance between the k -spheres centered at x ′ and y ′ ,and of radius | x ′′ | and | y ′′ | respectively. We denote by B H the balls for the distance d H , i.e., B H ( x, r ) = { y ∈ R N : d H ( x, y ) < r } . Lemma 4.4.
Suppose that the assumptions of Theorem 4 are satisfied. Let u ε ∈ H V, H ( R N ) be positive solutions of ( P ε ) found in Theorem 4, ( ε n ) n ⊂ R + and ( x n ) n ⊂ R n be sequences such that ε n → and x n = ( x ′ n , x ′′ n ) → ¯ x = (¯ x ′ , ¯ x ′′ ) ∈ ¯Λ as n → ∞ . If lim inf n →∞ u ε n ( x n ) > , (28) then we have, up to a subsequence, lim inf R →∞ lim inf n →∞ ε − ( N − k ) n Z T n ( R ) (cid:16) ε n |∇ u ε n | + V u ε n (cid:17) − G ε n ( x, u ε n ) ! ≥ ω k M (¯ x ) , where T n ( R ) := B H ( x n , ε n R ) .Proof. Let v n be defined by (23). Passing to a subsequence if necessary, we mayassume that there exists v ∈ H ( R N − k ) such that v n → v in C ( R N − k ). Since Λ issmooth, we can also assume that the sequence of characteristic functions χ n ( y, z ) = χ Λ ( x ′ n + ε n y, | x ′′ n | + ε n z ) converges almost everywhere to a measurable function χ satisfying 0 ≤ χ ≤
1. We then deduce that v solves the limiting equation − ∆ v + V (¯ x ) v = ˜ g ( y, v ) in R N − k , where ˜ g ( y, s ) := χ ( y ) K (¯ x ) f ( s ) + (1 − χ ( y )) min { µV (¯ x ) s, K (¯ x ) f ( s ) } . By (28), we know that v (0) = lim n →∞ v n (0) >
0, so that v is not identically zero. ONLINEAR SCHR ¨ODINGER EQUATION 13
It was shown in [8, Lemma 14] thatlim inf R →∞ lim inf n →∞ Z B (0 ,R ) (cid:18) (cid:16) |∇ v n ( y, z ) | + V ( x ′ n + ε n y, | x ′′ n | + ε n z ) | v n ( y, z ) | (cid:17) − G ε n ( x ′ n + ε n y, | x ′′ n | + ε n z, v n ( y, z )) (cid:19) dz dy ≥ Z R N − k (cid:16) |∇ v | + V (¯ x ) v (cid:17) − Z R N − k ˜ G ( y, v ( y )) dy, where ˜ G ( x, s ) := R s ˜ g ( x, σ ) dσ .Set B n ( R ) := B (( x ′ , | x ′′ | ) , ε n R ) ⊂ R N − k . By a computation similar to the oneleading to (22), we have Z T n ( R ) (cid:18) (cid:16) ε n |∇ u ε n ( x ) | + V ( x ) | u ε n ( x ) | (cid:17) − G ε n ( x, u ε n ( x )) (cid:19) dx = ω k Z B n ( R ) (cid:18) (cid:16) ε n |∇ ˜ u ε n ( x ′ , r ) | + V ( x ′ , r ) | ˜ u ε n ( x ′ , r ) | (cid:17) − G ε n ( x ′ , r, ˜ u ε n ( x ′ , r )) (cid:19) r k dr dx ′ = ω k | ¯ x ′′ | k ε N − kn Z B (0 ,R ) (cid:18) (cid:16) |∇ v n ( y, z ) | + V ( x ′ n + ε n y, | x ′′ n | + ε n z ) | v n ( y, z ) | (cid:17) − G ε n ( x ′ n + ε n y, | x ′′ n | + ε n z, v n ( y, z )) (cid:19) dz dy + o (1) . The conclusion follows. (cid:3)
Lemma 4.5.
Suppose that the assumptions of Theorem 4 are satisfied. Let u ε ∈ H V, H ( R N ) be positive solutions of ( P ε ) found in Theorem 4, ( ε n ) n ⊂ R + and ( x in ) n ⊂ R N be sequences such that ε n → and for ≤ i ≤ M , x in → ¯ x i ∈ ¯Λ as n → ∞ . Then, up to a subsequence, we have lim inf R →∞ lim inf n →∞ ε − ( N − k ) n Z R N \T n ( R ) (cid:16) ε n |∇ u ε n | + V u ε n (cid:17) − G ε n ( x, u ε n ) ! ≥ , where T n ( R ) := S Ki =1 B H ( x in , ε n R ) .Proof. See [8, Lemma 15]. (cid:3)
Proposition 4.6 (Lower estimate of the critical value) . Suppose that the assump-tions of Theorem 4 are satisfied. Let u ε ∈ H V, H ( R N ) be positive solutions of ( P ε ) found in Theorem 4, ( ε n ) n ⊂ R + and ( x in ) n ⊂ R N be sequences such that ε n → and for ≤ i ≤ M , x in → ¯ x i ∈ ¯Λ as n → ∞ . If for every ≤ i < j ≤ M , we have lim sup n →∞ d H ( x in , x jn ) ε n = ∞ and if for every ≤ i ≤ M , lim inf n →∞ u ε n ( x in ) > , then the critical value c ε defined in (21) satisfies lim inf n →∞ ε − ( N − k ) n c ε n ≥ ω k M X i =1 M (¯ x i ) . Proof.
This is a consequence of the two previous lemmas, see [8, Proposition 16]for the details. (cid:3)
The following proposition is the key result for the next section.
Proposition 4.7 (Uniform convergence to 0 outside small balls) . Suppose thatthe assumptions of Theorem 4 are satisfied and that Λ satisfies the assumptionsof Section 2.4. Let ( u ε ) ε ⊂ H V, H ( R N ) be positive solutions of ( P ε ) obtained inTheorem 4. If ( x ε ) ε> ⊂ Λ is such that lim inf ε → u ε ( x ε ) > , then(i) lim ε → M ( x ε ) = inf Λ ∩H ⊥ M ,(ii) lim ε → x ε , H ⊥ ) ε = 0 ,(iii) lim inf ε → d H ( x ε , ∂ Λ) > ,(iv) for every δ > , there exists ε > and R > such that, for every ε ∈ (0 , ε ) , k u ε k L ∞ (Λ \ B H ( x ε ,εR )) ≤ δ. Proof.
The first assertion is a direct consequence of Propositions 4.1 and 4.6, see[8, Proposition 33] for the details.For the second assertion, since ¯Λ is compact, we can assume by contradictionthat there exist sequences ( ε n ) n ⊂ R + and ( x n ) n ⊂ R N such that ε n → n →∞ u ε n ( x n ) > , and x n → ¯ x ∈ ¯Λ \ H ⊥ . If k = N −
2, let R ∈ O ( N ) denote the reflexion with respect to H ⊥ . Bydefinition of H V, H ( R N ), u ◦ R = u , and thuslim inf n →∞ u ε n ( R ( x n )) > . Since d H (¯ x, R (¯ x )) >
0, one has lim n →∞ d H ( x n ,R ( x n )) ε n = ∞ . By Proposition 4.6,we obtain lim inf n →∞ ε − ( N − k ) n c ε n ≥ ω k ( M (¯ x ) + M ( R (¯ x ))) ≥ ω k inf Λ M . which, together with Proposition 4.1lim inf n →∞ ε − ( N − k ) n c ε n ≤ ω k inf Λ ∩H ⊥ M is in contradiction with (10).In the case where k < N −
2, since inf Λ M >
0, choose ℓ ∈ N such that(29) inf Λ ∩H ⊥ M < ℓ inf Λ M . There exist isometries R , . . . , R l of R N such that R i ( H ) = H and R i (¯ x ) = R j (¯ x ),for every i, j ∈ { , . . . , ℓ } with i = j . One has hencelim n →∞ d H ( R i ( x n ) , R j ( x n )) ε = 0By Proposition 4.6, we getlim inf n →∞ ε − ( N − k ) n c ε n ≥ ω k ℓ X i =1 M ( R i (¯ x )) ≥ lω k inf Λ M , so that, in view of the upper estimate of Proposition 4.1, we have a contradictionwith (29). ONLINEAR SCHR ¨ODINGER EQUATION 15
For the third assertion, suppose by contradiction that there exist sequences( ε n ) n ⊂ R + and ( x n ) n ⊂ R N such that ε n → n →∞ u ε n ( x n ) > , and, x n → ¯ x ∈ ∂ Λ. We have just proven that ¯ x ∈ H ⊥ . By Proposition 4.6, we havelim inf n →∞ ε − ( N − k ) n c ε n ≥ ω k M (¯ x ) ≥ ω k inf ∂ Λ ∩H ⊥ M . This inequality, along with Proposition 4.1, contradicts (9).In order to obtain the last assertion, suppose by contradiction that there existsequences ( ε n ) n ⊂ R + , ( x n ) n and ( y n ) n ⊂ Λ such that ε n → u ε n ( y n ) ≥ δ, and lim n →∞ d H ( x n , y n ) ε n = ∞ . Up to a subsequence, we can assume that x n → ¯ x ∈ Λ and y n → ¯ y ∈ Λ. In view ofthe second assertion, one has ¯ x ∈ H ⊥ and ¯ y ∈ H ⊥ . Therefore, by Proposition 4.6,lim inf n →∞ ε − ( N − k ) n c ε n ≥ ω k ( M (¯ x ) + M (¯ y )) ≥ ω k inf Λ ∩H ⊥ M . In view of the assumption of (9), this would contradict Proposition 4.1. (cid:3) Barrier functions
Linear inequation outside small balls.
In this section we prove that for ε small enough, the solutions of the penalized problem ( P ε ) are also solutions ofthe initial problem (2). We follow the arguments of [16]. First we notice that thesolutions of ( P ε ) satisfy a linear inequation outside small balls. Lemma 5.1.
Suppose that the assumptions of Proposition 4.7 are satisfied andlet ( u ε ) ε> ⊂ H V, H ( R N ) be positive solutions of ( P ε ) found in Theorem 4 and ( x ε ) ε> ⊂ Λ be such that lim inf ε → u ε ( x ε ) > . Then there exist ρ > and ε > such that for all ε ∈ (0 , ε ) , − ε (∆ u ε + Hu ε ) + (1 − µ ) V u ε ≤ in R N \ B H ( x ε , εR ) . (30) Proof.
Set η := inf x ∈ Λ µV ( x ) K ( x ) . Since V and K are bounded positive continuous functions on ¯Λ, η >
0. By ( f ),there exists δ > f ( s ) s ≤ η for all s ≤ δ. By Proposition 4.7, we can find ε > ρ > ε ∈ (0 , ε ], onehas u ε ( x ) ≤ δ for all x ∈ Λ \ B H ( x ε , ερ ) . Hence K ( x ) f ( u ε ( x )) ≤ µV ( x ) u ε ( x ) in Λ \ B H ( x ε , ερ ) . We conclude that − ε ∆ u ε + (1 − µ ) V u ε ≤ − ε ∆ u ε + V u ε − Kf ( u ε ) = 0 in Λ \ B H ( x ε , ερ ) . The fact that u ε satisfies (30) in R N \ Λ follows directly from the definition of thepenalized nonlinearity. (cid:3)
This lemma suggests that we can compare the solution u ε with supersolutionsof the operator − ε (∆ + H ) + (1 − µ ) V in order to obtain decay estimates of u ε .5.2. Comparison functions.
The next lemma provides a minimal positive solu-tions of the operator − ∆ − H in R N \ ¯Λ. Lemma 5.2.
For every ε > , there exists Ψ ε ∈ C (cid:0) ( R N \ { } ) \ Λ (cid:1) such that ( − ε (∆Ψ ε + H Ψ ε ) + (1 − µ ) V Ψ ε = 0 in R N \ ¯Λ , Ψ ε = 1 on ∂ Λ , and (31) Z R N \ Λ (cid:18) |∇ Ψ ε ( x ) | + | Ψ ε ( x ) | | x | (cid:19) dx < ∞ . Moreover, there exists
C > such that, for every x ∈ R N \ Λ and every ε > , (32) 0 < Ψ ε ( x ) ≤ C (1 + | x | ) N − . Proof.
The function Ψ ε is obtained by minimimizing Z R N \ Λ (cid:16) ε (cid:16) |∇ u | − Hu (cid:17) + (1 − µ ) V u (cid:17) dx on the set { u ∈ H V ( R N ) : u = 1 on ∂ Λ } . By classical elliptic regularity theory, Ψ ε ∈ C (cid:0) ( R N \ { } ) \ Λ (cid:1) . The estimate (31)follows from (11).In order to obtain the estimate (32) consider the problem ( − ∆Ψ − H Ψ = 0 in R N \ ¯Λ , Ψ = 1 on ∂ Λ . We have just proved that this problem has a solution Ψ ∈ C (( R N \ { } ) \ Λ) suchthat(33) Z R N \ Λ (cid:18) |∇ Ψ( x ) | + | Ψ( x ) | | x | (cid:19) dx < ∞ . Now set for ρ ∈ (0 ,
1) and x ∈ B (0 , ρ ), W ( x ) := ( N − β − κ (cid:18) log 1 | x | (cid:19) − β , We compute that − ∆ W ( x ) = κβ | x | " ( N − (cid:18) log 1 | x | (cid:19) − (1+ β ) + (1 + β ) (cid:18) log 1 | x | (cid:19) − (2+ β ) . Since for | x | ≤ H ( x ) ≤ κ ( | x | log | x | ) β the function W is a supersolution of − ∆ − H in B (0 , ρ < N − β (cid:18) log 1 ρ (cid:19) β > κ, ONLINEAR SCHR ¨ODINGER EQUATION 17 W is positive on ∂B (0 , ρ ). In view of (33) Proposition 3.1 implies that Ψ is boundedfrom above by a positive multiple of W in B (0 , ρ ). Since Ψ is continuous and W isbounded in B (0 , B (0 , W ( x ) := 1 | x | N − (cid:16) ( N − β − κ (log | x | ) − β (cid:17) (see Lemma 3.4 of [16]), we obtain that Ψ( x ) ∼ | x | N − . We have thus proven thatΨ( x ) ≤ C (1 + | x | ) N − . Now, note that since V is nonnegative, − ∆Ψ ε − H Ψ ε ≤ . In view of (33) and (31), Proposition 3.1 is applicable, and for every x ∈ R N \ Λ,Ψ ε ( x ) ≤ Ψ( x ) ≤ C (1 + | x | ) N − . (cid:3) As explained in [16], the estimate (32) is the best one can hope for if V decaysrapidly at infinity or is compactly supported. However, if V decays quadraticallyor subquadratically at infinity, we can improve (32). Lemma 5.3.
Let Ψ ε be given by Lemma 5.2.(1) If lim inf | x |→∞ V ( x ) | x | > , then there exist λ > , R > and C > suchthat for every ε > and x ∈ R N \ B (0 , R ) , Ψ ε ( x ) ≤ C (cid:18) R | x | (cid:19) N − + q ( N − ) − κ + λ ε . (2) If lim inf | x |→∞ V ( x ) | x | α > with α < , then there exist λ > , R > , C > and ε > such that for every ε ∈ (0 , ε ) and x ∈ R N \ B (0 , R ) , Ψ ε ( x ) ≤ C exp (cid:18) − λε (cid:16) | x | − α − R − α (cid:17)(cid:19) . (3) If lim inf | x |→ V ( x ) | x | > , then there exist λ > , r > and C > such thatfor every ε > and x ∈ B (0 , r ) , Ψ ε ( x ) ≤ C (cid:18) | x | r (cid:19) q ( N − ) − κ + λ ε − N − . (4) If lim inf | x |→ V ( x ) | x | α > with α < , then there exist λ > , R > , C > and ε > such that for every ε ∈ (0 , ε ) and x ∈ B (0 , r ) , Ψ ε ( x ) ≤ C exp (cid:18) − λε (cid:16) | x | − α − − r − α − (cid:17)(cid:19) . Proof.
For (1), there exist
R > λ > x ∈ R N \ B (0 , R )(1 − µ ) V ( x ) ≥ λ | x | . One then checks that W ( x ) = (cid:18) R | x | (cid:19) N − + q ( N − ) − κ + λ ε is a supersolution in R N \ B (0 , R ). For (2), there exist
R > η > x ∈ R N \ B (0 , R )(1 − µ ) V ( x ) ≥ η | x | α . One then checks that W ( x ) = exp (cid:18) − λε (cid:16) | x | − α − R − α (cid:17)(cid:19) is a supersolution in R N \ B (0 , R ) with λ < ( − α ) ν and ε small enough.The proofs of the other assertions are similar. (cid:3) The other tool is a function that describes the exponential decay of u ε inside Λ. Lemma 5.4.
Let ¯ x ∈ Λ and R > be such that (34) B H (¯ x, R ) ⊂ Λ . Define Φ ¯ xε ( x ) := cosh (cid:18) λ R − d H ( x, ¯ x ) ε (cid:19) . (35) There exists λ > and ε > such that for every ε ∈ (0 , ε ) , one has − ε ∆Φ ¯ xε + (1 − µ ) V Φ ¯ xε ≥ in B H (¯ x, R ) . Proof.
First one computes − ε ∆Φ ¯ xε ( x ) = − λ cosh (cid:18) λε ( R − d H ( x, ¯ x )) (cid:19) + ελd H ( x, ¯ x ) (cid:18) N − − k | ¯ x ′′ || x ′′ | (cid:19) sinh (cid:18) λε ( R − d H ( x, ¯ x )) (cid:19) . Let us choose λ > λ < (1 − µ ) inf Λ V . In view of (34), one has for x ∈ B H (¯ x, R ), − ε ∆Φ ¯ xε ( x ) + (1 − µ ) V Φ ¯ xε ( x ) ≥ ελd H ( x, ¯ x ) (cid:18) N − − k | ¯ x ′′ || x ′′ | (cid:19) sinh (cid:18) λε ( R − d H ( x, ¯ x )) (cid:19) + (cid:16) (1 − µ ) inf Λ V − λ (cid:17) cosh (cid:18) λε ( R − d H ( x, ¯ x )) (cid:19) . This last expression is positive if ε is sufficiently small. (cid:3) Lemma 5.5.
Let ( x ε ) ε ⊂ Λ be such that lim inf ε → d H ( x ε , ∂ Λ) > and ρ > . Then, there exist ε > and a family of functions ( W ε ) <ε<ε ⊂ C , (( R N \ { } ) \ B H ( x ε , ερ )) such that for all ε ∈ (0 , ε ) , one has(i) W ε satisfies the inequation − ε (∆ + H ) W ε + (1 − µ ) V W ε ≥ in R N \ B H ( x ε , ερ ) , (ii) ∇ W ε ∈ L ( R N \ B H ( x ε , ερ )) and W ε | x | ∈ L ( R N \ B H ( x ε , ερ )) ,(iii) W ε ≥ on ∂B H ( x ε , ερ ) ,(iv) for every x ∈ B H ( x ε , ερ ) , W ε ( x ) ≤ C exp (cid:18) − λε d H ( x, x ε )1 + d H ( x, x ε ) (cid:19) (1 + | x | ) − ( N − , x ∈ R N . Moreover,
ONLINEAR SCHR ¨ODINGER EQUATION 19 (1) If lim inf | x |→∞ V ( x ) | x | > , then there exists λ > , ν > and C > suchthat for ε > small enough, W ε ( x ) ≤ C exp (cid:18) − λε d H ( x, x ε )1 + d H ( x, x ε ) (cid:19) (1 + | x | ) − νε . (2) If lim inf | x |→∞ V ( x ) | x | α > with α > , then there exists λ > and C > such that for ε > small enough, W ε ( x ) ≤ C exp (cid:18) − λε d H ( x, x ε )1 + d H ( x, x ε ) (1 + | x | ) − α (cid:19) . (3) If lim inf | x |→ V ( x ) | x | > , then there exists λ > , ν > and C > suchthat for ε > small enough, W ε ( x ) ≤ C exp (cid:18) − λε d H ( x, x ε )1 + d H ( x, x ε ) (cid:19) (cid:18) | x | | x | (cid:19) νε . (4) If lim inf | x |→ V ( x ) | x | α > with α > , then there exists λ > and C > such that for ε > small enough, W ε ( x ) ≤ C exp − λε d H ( x, x ε )1 + d H ( x, x ε ) (cid:18) | x | | x | (cid:19) α − ! . Proof.
Let Ψ ε be given by Lemma 5.2. Choose a set U ⊂ R N such that ¯Λ ⊂ U ,0 ¯ U and ¯ U is compact. Choose ˜Ψ ε ∈ C ( R N \ { } ) ∩ H ( R N ) such that ˜Ψ ε = Ψ ε in R N \ U and ˜Ψ ε = 1 in Λ. In view of the estimate of Lemma 5.2, one can alsoensure that sup ε> (cid:13)(cid:13)(cid:13) ˜Ψ ε (cid:13)(cid:13)(cid:13) L ∞ ( U ) < ∞ . Choose R >
R < lim inf ε → dist( x ε , ∂ Λ) . Let Φ x ε ε be given by (35) and set w ε ( x ) := (cid:26) Φ x ε ε ( x ) if x ∈ B H ( x ε , R ) , ˜Ψ ε ( x ) if x ∈ R N \ B H ( x ε , R ) . By (36), for ε small enough, B H ( x ε , R ) ⊂ Λ so that w ε ∈ C , ( R N ). Moreover, if ε is small enough, Lemma 5.4 is applicable and in B H ( x ε , R ) \ B H ( x ε , ερ ), we have − ε (∆ + H ) w ε + (1 − µ ) V w ε ≥ − ε ∆Φ x ε ε + (1 − µ ) V Φ x ε ε ≥ . In Λ \ B H ( x ε , R ), one has − ε (∆ + H ) w ε + (1 − µ ) V w ε = − ε H + (1 − µ ) (cid:16) inf Λ V (cid:17) ≥ , for ε small enough. In U \ Λ, one has − ε (∆ + H ) w ε + (1 − µ ) V w ε = − ε (∆ + H ) ˜Ψ ε + (1 − µ ) V ˜Ψ ε ≥ , for ε small enough since V ˜Ψ ε is positive on U . Finally, in R N \ U , one has − ε (∆ + H ) w ε + (1 − µ ) V w ε = − ε (∆ + H ) Ψ ε + (1 − µ ) V Ψ ε = 0 . We set W ε ( x ) := w ε ( x )cosh λ (cid:0) Rε − ρ (cid:1) , where λ is chosen as in the previous lemma. It is standard to see that W ε satis-fies properties (ii) and (iii). Statement (iv) follows from Lemma 5.2. The otherconclusions follow from Lemma 5.3. (cid:3) Thanks to the previous lemma, we obtain an upper bound on the solutions( u ε ) ε> of ( P ε ). Proposition 5.6.
Suppose that the assumptions of Proposition 4.7 are satisfied.Let ( u ε ) ε> ⊂ H V, H ( R N ) be the positive solutions of ( P ε ) found in Theorem 4 and ( x ε ) ε> ⊂ Λ be such that lim inf ε → u ε ( x ε ) > . Then there exist
C > , λ > and ε > such that for all ε ∈ (0 , ε ) , u ε ( x ) ≤ C exp (cid:18) − λε d ( x, S kε )1 + d ( x, S kε ) (cid:19) (1 + | x | ) − ( N − , x ∈ R N . (37) Moreover, (1) , (2) , (3) and (4) in Lemma 5.5 hold with u ε in place of W ε .Proof. By Lemma 5.1, there exist ρ > ε > ε ∈ (0 , ε ),the solution u ε satisfies inequation (30). Further, k u ε k L ∞ ( B H ( x ε ,ερ )) is bounded as ε → W ε ) ε be the family of barrier functions given byLemma 5.5. By Proposition 3.1, we have u ε ( x ) ≤ k u ε k L ∞ ( B H ( x ε ,ερ )) W ε ( x ) in R N \ B H ( x ε , ερ ) , and the conclusion comes from Lemma 5.5. (cid:3) We are now in a position to prove Theorem 3.
Proof of Theorem 3.
We know from Theorem 4 that the modified equation ( P ε )possesses a positive solution u ε ∈ H V, H ( R N ). In order to prove that for ε smallenough, this solution actually solves (2), it suffices to show that, for every x ∈ ( R N \ { } ) \ Λ, one has K ( x ) f ( u ε ( x )) u ε ( x ) ≤ ε H ( x ) + µV ( x ) . Assume that V and K satisfy ( G ∞ ) and ( G ), by Proposition 5.6 and assumptions( f ) and ( f ), if ε > x ∈ R N \ Λ, K ( x ) f ( u ε ( x )) u ε ( x ) ≤ K ( x ) f (cid:16) Ce − λε (1 + | x | ) − ( N − (cid:17) Ce − λε (1 + | x | ) − ( N − ≤ Ce − λε ( q − (1 + | x | ) σ − ( N − q − ≤ ε κ | x | ((log | x | ) + 1) β = ε H ( x ) . The other cases can be treated in a similar way. (cid:3)
In some settings, it is interesting to determine whether the solutions are in L .We obtain as a byproduct the following Corollary 5.7.
Let u ε be the solution of (2) found in Theorem 3. If N ≥ or lim inf | x |→∞ | x | V ( x ) > , then, for ε small enough, u ε ∈ L ( R N ) .Proof. This follows immediately from Proposition 5.6. (cid:3) The two-dimensional case
In dimension N = 2, the method has to be modified because the classical Hardyinequality fails on unbounded domains of R . Let us recall the Hardy-type inequal-ity that was proved in [16, Lemma 6.1]: ONLINEAR SCHR ¨ODINGER EQUATION 21
Lemma 6.1.
Let
R > r . Then there exists
C > such that for every u ∈ D ( R ) , Z R |∇ u | + C Z B (0 ,R ) \ B (0 ,r ) u ≥ Z R \ B (0 ,R ) u ( x ) | x | (cid:16) log | x | r (cid:17) dx. We deduce therefrom
Lemma 6.2. If V ∈ C ( R \ { } ) is nonnnegative and non identically , then thereexists κ > such that for ε > sufficiently small, for every u ∈ D ( R ) , κ Z R u ( x ) | x | (cid:16) | x | ) (cid:17) dx ≤ Z R ε |∇ u | + V u . Proof.
One sees that by the conformal transformation x x | x | , Lemma 6.1 be-comes Z R |∇ u | + C Z B (0 ,R ) \ B (0 ,r ) u ≥ Z B (0 ,r ) u ( x ) | x | (log | x | ) dx. Therefore, one has Z R u ( x ) | x | (cid:16) | x | ) (cid:17) dx ≤ C (cid:16)Z R |∇ u | + Z B (0 , \ B (0 , / u (cid:17) . Since V is continuous and does not vanish identically, there exists ¯ x ∈ R and¯ r >
0, such that inf B (¯ x, ¯ r ) V >
0. Hence, there exists
C > Z B (0 , \ B (0 , / u ≤ C (cid:16)Z R |∇ u | + V | u | (cid:17) . Bringing the inequalities together, there exists
C > Z R u ( x ) | x | (cid:0) | x | ) (cid:1) dx ≤ C Z R ( |∇ u | + V u ) . This brings the conclusion when ε > (cid:3)
The space H V ( R ) can thus be defined as in the case N > D ( R ) with respect to the norm defined by (13).The penalization potential H : R → R is defined by H ( x ) := κ | x | (1 + (log | x | ) ) β . where β > κ ∈ (0 , κ ) We see that H ( x ) ≤ κ | x | (1 + (log | x | ) ) . Together with Lemma (6.2), this ensures positivity of the quadratic form associatedto − ε (∆ + H ) + V .As in the case N >
2, this inequality implies the following comparison principle.
Proposition 6.3.
Let Ω ⊂ R be a smooth domain. Let v, w ∈ H (Ω) ∩ C (Ω) besuch that ∇ ( w − v ) − ∈ L (Ω) , ( w − v ) − / ( | x | (1 + | log | x || )) ∈ L (Ω) and − ε (∆ + H ) w + V w ≥ − ε (∆ + H ) v + V v, in Ω . If ∂ Ω = ∅ , assume also that w ≥ v on ∂ Ω . Then w ≥ v in Ω . One continues the proof of Theorem 3 as in the case N ≥
3. In Proposition 3.2,one takes A λ := B (0 , e e λ ) \ B (0 , e − e λ ) and η λ ( x ) := ζ (cid:18) log | log | x || λ (cid:19) . One then obtains estimate (19) by using Lemma 6.2 instead of Hardy’s inequality.The only other notable difference lies in the choice of the function W in the proofof Lemma 5.2, where one follows the construction of [16, Lemma 6.3], i.e. W ( x ) = β ( β + 1) − κ | log | x || − β . References [1] Antonio Ambrosetti and Andrea Malchiodi,
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The principle of symmetric criticality , Comm. Math. Phys. (1979),no. 1, 19–30. MR547524 (81c:58026) D´epartement de Math´ematique, Universit´e libre de Bruxelles, CP 214, Boulevarddu Triomphe, 1050 Bruxelles, Belgium
E-mail address : [email protected] D´epartement de Math´ematique, Universit´e cahtolique de Louvain, Chemin du Cy-clotron 2, 1348 Louvain-la-Neuve, Belgium
ONLINEAR SCHR ¨ODINGER EQUATION 23
D´epartement de Math´ematique, Universit´e libre de Bruxelles, CP 214, Boulevarddu Triomphe, 1050 Bruxelles, Belgium
E-mail address : [email protected]
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