Soliton solutions to systems of coupled Schrodinger equations of Hamiltonian type

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1 Soliton solutions to systems of couled Schrodinger euations of Hamiltonian tye Boyan Sirakov, Sérgio Soares To cite this version: Boyan Sirakov, Sérgio Soares. Soliton solutions to systems of couled Schrodinger euations of Hamiltonian tye. Transactions of the American Mathematical Society, American Mathematical Society, 2010, 362 (11), <hal v2> HA Id: hal htts://hal.archives-ouvertes.fr/hal v2 Submitted on 9 Jan 2009 HA is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. archive ouverte luridiscilinaire HA, est destinée au déôt et à la diffusion de documents scientifiues de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 Soliton solutions to systems of couled Schrödinger euations of Hamiltonian tye Boyan SIRAKOV 1 UFR SEGMI, Université Paris 10, Nanterre Cedex, France and CAMS, EHESS, 54 bd Rasail, Paris Cedex 06, France Sérgio H. M. SOARES 2 Deartamento de Matemática, Instituto de Ciências Matemáticas e de Comutação, Universidade de São Paulo, , São Carlos-SP, Brazil 1 Introduction A major role in uantum hysics is layed by the nonlinear Schrödinger euation i ψ t = 2 2m xψ + V (x)ψ f(x, ψ), (1.1) where m and are ositive constants, the wave ψ : R + R N C, N 3, V is a otential which is bounded below, and f = f(x, ψ )ψ is a nonlinear function, for instance in the classical cubic aroximation f = ψ 2 ψ. One of the uestions to which huge attention has been given during the last twenty years is the existence of stationary states (see (1.2) below) for small values of, which aear due to the geometry of the otential. This aer is devoted to the corresonding uestion of existence of solutions of some systems of Schrödinger euations. Systems of nonlinear Schrödinger tye have been widely used in the alied sciences but mathematical study of standing wave solutions was undertaken only very recently, romted in articular by the discovery of the imortance of these systems as models in nonlinear otics (see for instance [4], [9], [26]) and in the study of Bose-Einstein condensates (see [26], [39]). As in the large majority of other aers on the subject we consider here systems of two euations. So we suose ψ is a vector function, ψ = (ψ 1, ψ 2 ), and satisfies a system of euations like (1.1), with f = ( f 1, f 2 ) and f k = j f kj(x, ψ 1, ψ 2 )ψ j. We will be interested in soliton (standing wave) solutions of these systems, that is, solutions in the form ψ j (t, x) = e i(e/ )t u j (x). (1.2) 1 Corresonding author, sirakov@ehess.fr ; S.H.M. Soares : monari@icmc.us.br 2 Research suort in art by FAPESP. 1

3 Substituting (1.2) into (1.1) and setting b(x) = V (x) E leads to the system of real ellitic artial differential euations (we write u = u 1, v = u 2 ) { (S ) 2 u + b(x)u = f 1 (x, u, v) in R N, 2 v + b(x)v = f 2 (x, u, v) in R N. We suose this system is in variational form, that is, it is the Euler-agrange system of some energy functional. This haens when f 1, f 2 are the derivatives of some function H(x, u, v). There are two tyes of such systems, agrangian - when f 1 = H u, f 2 = H v (the above mentioned examles are of this tye), and Hamiltonian - when f 1 = H v, f 2 = H u. The simlest examle of a Hamiltonian system is the widely studied ane-emden system - when f 1 = v, f 2 = u in (S ). Even for this system imortant oen uestions subsist (see [15]). Hamiltonian systems are very usual in biology, more secifically in models in oulation dynamics (see [27]) whose stationary states verify systems of tye (S ), for instance with f 1 = v(u 2 + g 1 (v)), f 2 = u(v 2 + g 2 (u)). An imortant difficulty in the study of Hamiltonian systems (as oosed to agrangian) is the fact that the energy functional is strongly indefinite, that is, its leading art is resectively coercive and anticoercive on infinitely dimensional subsaces of the energy sace - we refer to [3] for a general discussion. The resent article is devoted to this case. Our goal is to get a general existence result for small in the case of a suerlinear and subcritical Hamiltonian system with a well otential. As in many alications, we consider traing (or well -tye) otentials, the standard examle being b(x) x x 0 2 in a neighbourhood of some x 0 R N. A articular case of our result will be the existence of soliton waves thanks to a global well structure of b, that is, 0 = inf b(x) < lim inf b(x). (1.3) x R N x Notice that inf x R N b(x) = 0 can always be achieved through the choice of E in (1.2). Unfortunately, as of today PDE theory lacks the means to tackle the existence uestion under hyothesis (1.3) only, even in the scalar case. However, it turns that we can show that (S ) has a solution rovided the constant is sufficiently small. Note that in ractice, the Planck constant, is a very small uantity, so it makes sense to study roblem (S ) at the limit 0. Here are the recise statements. We assume H(x, u, v) is differentiable and strictly convex in (u, v) R 2 for all x R N, H(x, 0, 0) = 0 and (H1) there exist constants,, α k, β k > 1, such that > N 2 N, α k β k = 1, (1.4)

4 and for some c 0, d 0 > 0, C k 0, D k 0 we have for x R N,(u, v) R 2 c 0 u H u (x, u, v) C 0 u + d 0 v H v (x, u, v) D 0 v + m C k u αk 1 v β k, k=1 m D k u α k v βk 1. (H2) There exists α > 2 such that for all x R N and (u, v) R 2 \ {(0, 0)} k=1 uh u (x, u, v) + vh v (x, u, v) αh(x, u, v) > 0. A tyical examle of a function satisfying these hyotheses is H(x, u, v) = a 0 (x) u + n 1 a i(x) u α i v β i + a n+1 (x) v +1, under (1.4). We suose that the continuous otential b(x) satisfies b 0 in R N and (b1) there exists x 0 R N (say x 0 = 0) such that b(x 0 ) = 0; (b2) there exists A > 0 such that the level set G A = {x R N : b(x) < A} has finite ebesgue measure. Note that the conditions (b1)-(b2) include (1.3) as a articular case. We shall also suose that b(x) is bounded. This condition is made for simlicity, since it is irrelevant to the goal of our aer, which is to use the well geometry of the otential. Actually it is even easier to consider otentials which are large at infinity (then there is no restriction on ), since the energy sace embeds comactly into ebesgue saces, see for instance Theorem 4 in [37]. Note also that (H1) means the roblem is suerlinear and subcritical, in other words, the coule (, ) is under the critical hyerbola (given by the ineuality (1.4)). In articular, one of the nonlinearities in (S ) can have growth larger than the exonent (N +2)/(N 2), rovided the growth of the other is smaller enough to comensate (note that when = (1.4) reduces to < (N + 2)/(N 2)). In this case the functional associated to (S ) is not defined for u, v H 1 (R N ). It is nowadays well-known that (1.4) is the right notion of subcriticality for a Hamiltonian system with ower-growth nonlinearity, see [7], [21], [35], [36]. The following theorem contains our main result. Theorem 1 If f 1 = H v, f 2 = H u, and (H1)-(H2), (b1)-(b2) are satisfied then (S ) has a nontrivial solution for small. 3

5 We now uote revious works related to this result. There is a huge literature for the scalar case we refer to [2], [5], [10], [14], [17], [19], [20], [23], [28], [29], [32], [38], [41] and to the references in these aers. Some tyes of agrangian systems with well otentials were studied in [1], [26], [30]. Existence results (for any ) for radially invariant Hamiltonian systems in R N were established in [16] and [37]. A result similar to Theorem 1 can be found in [33] (see also [34]) in the articular case when H = F (u) + G(v), that is, the right-hand side of the system is indeendent of x and has no cross-terms in u, v. This restrictive hyothesis is due to the method used in these aers, which extends to systems the arguments in [14]. Finally, in the recent aer [11] a fairly general result was roved on system (S ), but under the hyothesis that both, are smaller than (or in some cases eual to) the scalar exonent (N + 2)/(N 2). The method in [11] is based on an alication of a linking theorem to the energy functional associated to (S ). The starting oint for our work is [38], where the scalar version of Theorem 1 was roved. The method in [38] extends readily to agrangian systems, since then the energy functional has the same geometry as the scalar one, but the situation aears to be considerably more involved for Hamiltonian systems. We have used a dual variational structure, relying on the egendre Fenchel transformation, which allows us to transform the roblem into a new one, to which the Mountain Pass Theorem (without the Palais- Smale condition) alies. However, then one of the key observations that the generalized mountain ass value tends to zero as 0 turns out to be rather delicate to rove, and the method of roof in [38] fails. We have found a way to deal with this roblem by Fourier analysis, a tool that is seldom encountered in this branch of the calculus of variations. Our method will hoefully be useful in other situations as well. So the main interest of Theorem 1 is twofold first, it extends and joins together revious existence results of this tye, giving an otimal range for the growth of the nonlinearities involved ; and second, its roof is based on a new idea, namely the use of Fourier transforms in the study of the behaviour of generalized critical values. We finally remark that in the scalar case it has recently been established that standing wave solutions of (1.1) can be shown to exist for nonlinearities which grow suercritically - see [6], [12], [13]. In the light of these results, we exect that our hyotheses on the growth of f 1, f 2 at infinity can be relaxed, at least for some tye of nonlinearities. The aer is organized as follows. The next section is reliminary - we describe the variational setting we use. The main frame of the roof of Theorem 1 is to be found in Section 3. Finally, the core result the fact that the mountain ass values (and hence the norms and the energy) of the 4

6 solutions we find tend to zero as 0 is roved in Section 4. 2 The dual variational formulation We start by recalling some facts which ermit us to set u the variational framework for solving system (S ). emma 2.1 et V be bounded and nonnegative function satisfying (b1) and (b2). Then, for every g s (R N ), 1 < s <, and > 0, the roblem u + V ( x)u = g in R N ossesses a uniue solution u W 2,s (R N ). In addition, there exits a constant K > 0 (which may deend of ) such that u W 2,s (R N ) K g s (R N ) Proof: Denote V (x) = V ( x). For s (1, ), consider the oerator R s : W 2,s (R N ) s (R) defined by R s u = ( + V I)u for u W 2,s (R N ). It follows for instance from Theorem 1 of [31] that (i) Ker (R s λi) = Ker (R 2 λi), for every s (1, ). (ii) s (R N ) = Ker (R s λi) Im (R s λi). Since V (R N ), it is known (see for examle emma 3.10 in [40]) that the sectrum σ(r 2 ) [Λ, ) and Λ σ(r 2 ), where { } Λ = inf ( u 2 + V (x)u 2 ) u H 1 (R N ), u 2 = 1. It follows from emma 1 in [38] that Λ > 0. Therefore 0 σ(r 2 ). Conseuently Ker (R s ) = Ker (R 2 ) = {0} and s (R N ) = Ker (R s ) + Im(R s ) = Im(R s ). Thus, R s : W 2,s (R N ) s (R N ) s (R N ) is a isomorhism. Note that R s is continuous thanks to the immersion W 2,s (R N ) s (R N ). So, there exists a ositive constant C such that for all u s (R N ) R 1 s u W 2,s (R N ) C u s (R N ). 5

7 1 Given, > 1 such that > N 2, we define the oerators N R : (R N 2, ) W (R N ), S : +1 (R N +1 2, ) W (R N ), by R = S = ( + b I) 1, where b (x) = b( x). It follows from emma 2.1 that the oerators R and S are well defined and continuous. Since 1/( + 1) > /( + 1) 2/N holds, we have the continuous Sobolev embeddings 2, i 1 : W (R N ) +1 (R N +1 2, ), i 2 : W (R N ) (R N ), conseuently R. = i1 R, S. = i2 S are linear continuous oerators. So we can define the linear oerator T : +1 (R N ) (R N ) +1 (R N ) (R N ), T := ( 0 R S 0 that is, for all f, φ +1 (R N ), g, ϕ (R N ), T w, η = φr g + ϕs f, η = (φ, ϕ), w = (f, g). et X = +1 (R N ) (R N ) be the Banach sace endowed with the norm ), w = f g 2 ; w = (f, g) X, from now on s and hdx will denote the s norm in R N and h(x)dx, R N resectively. The dual functional Ψ : X IR is defined by Ψ (w) = H (x, w) dx 1 T w, w dx, w X, 2 where H is the egendre-fenchel transform of H, that is, for all x R and w = (w 1, w 2 ) R 2, H (x, w) = su t R 2 {w 1 t 1 + w 2 t 2 H(x, t)}. emma 2.2 The functional Ψ is well defined and C 1 on X. Its Fréchet derivative is given by (Ψ ) (w)η = Hw(x, w)η dx T w, η dx, η X. 6

8 If w = (f, g) is a critical oint of Ψ, then (u, v) = T h w is a solution of the system (obtained by (S ) through the change x x) { u + b(hx)u = Hv (hx, u, v) in R N, v + b(hx)v = H u (hx, u, v) in R N (S. ) Proof: The roof of this lemma is known, for instance we can emloy the arguments given in [8] (see emma 4.3 there, and also [22]). et us sketch it for comleteness. The derivative of the second term in Ψ is simle to get, by the relation η, T w dx = w, T η dx, η, w X. Consider the functional H(z) = H(x, z) dx, H : X = +1 (R N ) (R N ) R, where z = (u, v). From the hyotheses on H it follows that H is well-defined on X and is a C 1 -functional. The egendre-fenchel transform of H is given by H (w) = H (x, w) dx, H : X R. Since H is strictly convex the gradient H z : R 2 R 2 is a homeomorhism. Thus, H is a bijection from X to X, which is continuous and bounded. Furthermore, H is Gâteaux differentiable, (H ) (w) = (H ) 1 (w) for every w X (this is a characterization of the egendre-fenchel transform), and (H ) (w)η = Hw(x, w)η dx, η, w X. Thus, (H ) : X X is continuous and bounded, which imlies that H is Fréchet differentiable. Now, if w is a critical oint of Ψ, it follows that z = (u, v) = T h w is a solution of (S ). In fact, we have (H ) (w) T h w = 0 in X, that is As a result, (H ) 1 (w) z = 0 in X. T 1 h z +1 (H 2, 2, )(z) = 0 in W W, 7

9 because T , 2, h is an isomorhism between W W and +1. Thus, (u, v) = z = T h w is a solution of system (S ). We say that w = (f, g) is the dual solution associated to (u, v). By making the change of variable x 1 x in R N, system (S ) becomes { 2 u + b(x)u = H v (x, u, v) in R N, 2 v + b(x)v = H u (x, u, v) in R N. (S ) 3 Proof of Theorem 1 We start with the following simle fact. emma 3.1 The functional Ψ has a mountain ass geometry on the sace X, in the sense that there exist ρ, α > 0 and w X such that Ψ Bρ α, Ψ (w) < 0 and w > ρ. Proof: It is easy to see that (H1) and (H2) imly that there exist ositive constants c 1 c 4 such that c 1 f +1 + c 2 g H(x, w) c 3 f +1 + c 4 g, w = (f, g). From roerties of egendre-fenchel transformations, we have d 1 f +1 + d 2 g H (x, w) d 3 f +1 + d 4 g, (3.5) for some ositive constants d 1 d 4. By using the Hölder ineuality and the boundedness of R and S, for all w = (f, g) X we easily get w, T w C( f +1 g C( g 2 Then, from (3.5) and (3.6) we get + g f +1 ) + f 2 +1 ) = C w 2 X, (3.6) Ψ +1 (w) C( f +1 + g ) C( f g 2 ). Thus, since ( + 1)/ < 2 and ( + 1)/ < 2, for each > 0 there exist constants ρ, α > 0 such that Ψ Bρ α. Now, we claim we can find w X such that Ψ (w) < 0 and w > ρ. In fact, there exists w + = (f +, g + ) X such that T h w +, w + > 0 (indeed, it 8

10 is sufficient to take f + = g + Cc (R N )). By using (3.5) we obtain, for all t > 0, Ψ (tw + ) Ct +1 f +1 + Ct g t2 T h w +, w +, 2 for some ositive constant C. Since sufficiently large. and Set, +1 < 2, the claim follows for t > 0 Γ. = {γ C([0, 1], X) : γ(0) = 0, Ψ (γ(1)) < 0} c = inf γ Γ max t [0,1] Ψ (γ(t)). Standard critical oint theory imlies that for each > 0 we can find a seuence {w n} n=1 X such that Ψ (w n) c and (Ψ ) (w n) 0 as n. (3.7) Our goal will be to show that for sufficiently small values of each of these seuences ossesses an accumulation oint, which is nontrivial solution of (S ). emma 3.2 For > 0 fixed, the seuence w n = (f n, g n) is bounded in X. Proof: From roerties of the egendre-fenchel transform and (H2) we have H (x, w n) (1 1 α )H f (x, w n)f n + (1 1 α )H g (x, w n)g n. (3.8) Now H (x, wn) = 1 T h w 2 n, wn + Ψ (wn) = Ψ (wn) 1 2 (Ψ ) (wn), wn + 1 H 2 w(x, wn)w n. Setting λ = α 2(α 1) < 1, from (3.7) and (3.8) we obtain (1 λ) H (x, wn) c + o n (1) wn X, (3.9) where o n (1) is a uantity which tends to zero as n. By combining (3.5) and (3.9) we get for some k, K > 0 k wn γ X f n gn 9 Kc + o n (1) w n X, (3.10)

11 with γ = min{1 + 1/, 1 + 1/} > 1. bounded in X, for > 0 fixed. This trivially imlies that {wn} is With the hel of emma 3.2 for each > 0 we can extract a subseuence of {w n} which converges weakly in X to a function w = (f, g ). We affirm that w is a critical oint of Ψ. First, for each > 0 the seuence z n = T h w n is clearly bounded in X, since T is bounded. Another way of writing (3.7) is T 1 h z n (H )(z n ) = o n (1) (see the roof of emma 2.2). Since u to a subseuence we have z n z in +1 2, 2, W W we see that the limit function z is a weak solution of (S ). This imlies that T h z X and w = T h z is a critical oint of Ψ. It remains to show that w is not identically zero. We claim that for small this is the case. The roof of this claim will be carried out through several stes. First, let u n and vn be the functions given by that is, u n = R gn 2, W (R N ) and vn = S fn +1 2, W (R N ), (3.11) u n + b( x)u n = g n and v n + b( x)v n = f n, x R N. (3.12) Next, we note that (1.4) ermits to us to choose s, t such that 0 < s, t < 2, s + t = 2 and t < N 2, 2 t < N 2, N( 1) 2( + 1) 4( + 1) N( 1) < t <. (3.13) 2( + 1) Then + 1 < 2N and + 1 < N 2t 2N, which imlies N 2s 2, W H s , and W H t, where H s, H t are the usual fractional Sobolev saces over R N. emma 3.3 There exists a constant β > 0 (indeendent of ) such that for each > 0 we can find R = R( ) > 0, for which u n +1 (+1) βc H s (R N ) + β u n +1 H s (B R ) + o n(1), vn () +1 βc H t (R N ) + β vn H t (B R ) + o n(1). 10

12 Proof: We shall need some functional analysis. For s (0, 1) let H s b( x) be the sace of the functions u such that b 1 2 ( x)u 2 (R N ) and u(x) u(y) x y s+ N 2 2 (R N R N ). One can also define Hb( x) s by interolation between the saces 2 b( x) = {u : b( x)u 2 < } and Hb( x) 1 = {u 2 b( x) : u 2 < }. Since b (R N ), the inclusion H s Hb( x) s holds. On the other hand it is standard to check that Hb( x) s (RN ) is embedded into H s (B R ), for any s > 0 and any ball B R. Once more through emma 1 in [38] (see also the argument used in the roof of this lemma) we can rove that H s (R N ) = Hb( x) s (RN ) under hyotheses (b1) and (b2). Define = + b( x) : H ( is a ositive oerator) and A s := ( ) s, so that A s : Hb( x) s 2 is a isomorhism between Hb( x) s and 2. This is standard functional analysis, for details and references see [16], ages , where the case b 1 was considered. We observe that u H s b( x) = A s u 2. Then the weak formulation of the first euation in (3.12) is A s u na t ϕ = gnϕ, ϕ H t. (3.14) Putting ϕ = A t A s u n into (3.14), we obtain u 2 H = A s u n 2 = b( x) s g na t A s u n. So there exists a ositive constant indeendent of such that for all R > 0 u n 2 H s b( x) (RN ) g n A t A s u n C g n = C g n u n H s [ ] u n H s (B R ) + u n H s (R N \B R ). (3.15) On the other hand, hyotheses (b1) and (b2) imly that we can find c > 0 such that for any > 0 there exists R = R( ) for which w H s b( x) (R N ) c w H s (R N \B R ), w H s b( x)(r N ) = H s (R N ). This ineuality (articularly easy to check under (1.3)) follows from emma 3 in [38] where the case s = 1 was studied, and from an interolation argument. 11

13 Since x 2 a + bx, x 0 imlies x C(b + a) we get from (3.15) u n H s (R N \B R ) C g n (R N ) + C u n H s (B R ), (3.16) for some ositive constant C indeendent of. Similarly, v n H t (R N \B R ) C f n +1 (R N ) + C v n H t (B R ). (3.17) Recall we already roved (emma 3.2 and (3.10)) that there exists a ositive constant C indeendent of for which g n Cc + o n (1) and fn Cc + o n (1). (3.18) By combining these with (3.16) and (3.17) we get emma 3.3. The final and basic ingredient of the roof of Theorem 1 is the following emma 3.4 We have lim c = 0. (3.19) 0 The roof of this lemma will be given in the next section. We shall now roceed to the roof of Theorem 1. Proof of Theorem 1. Since H (w) is a convex function on R 2 we have H (w), w H (w) for all w R 2. Hence [ c = lim Ψ (w n n) (Ψ ) (wn), wn ] 1 2 lim n lim su n ( T w n, w n = 1 2 lim n f n +1 f nr g n + g ns f n R g n +1 + g n S f n By the Hölder ineuality for each ε > 0 there exists C = C(ε) > 0 such that ). c ε lim su n ( fn gn ) + C lim su( R gn +1 + S n +1 fn ), so by using (3.18) and by choosing ε sufficiently small we get by the Sobolev embedding and the boundedness of R, S that ( ) c C lim su R gn +1 + S n +1 fn C lim su u n +1 H n s (R N ) 12 + C lim su vn. H n t (R N )

14 Therefore, by the revious lemma, ( ) (+1) () +1 c β c + c + C lim su u n +1 H s (B R ) n + C lim su vn H t (B R ). n 2, Note the embeddings W H s +1 2,, W H t are comact on bounded domains, so {u n}, {vn} converge strongly on B R as n. Hence for the limit functions u, v we get u +1 H s (B R ) + v H t (B R ) [1 C 1 β ( c 1 + c 1 +1 )]. However the last uantity is strictly ositive for small (since c 0), which means that the limit functions are not identically zero. Note that of course (f, g ) = (0, 0) if and only if (u, v ) = (0, 0). 4 Proof of emma 3.4. We start by observing that c inf su Ψ (tw) = inf max w X\{0} w E t 0 Ψ (tw), t 0 where we have set E = {w X T w, w > 0}. An exlicit comutation (see Aendix I) shows that for any w E we have ( w 2 γ max t 0 Ψ (tw) const., (4.20) T w, w ) where γ = max { 2, } So to rove emma 3.4 it will be enough to establish the following claim: inf w E w 2 T w, w In order to verify this, we observe that inf w E w 2 T w, w = 0, as 0. (4.21) inf w E : w =1 Thus, (4.21) is euivalent to the following result T w, w.

15 emma 4.1 We have su w E : w =1 T w, w dx + as 0. (4.22) To facilitate the task of the reader, we first describe the idea behind the roof of (4.22). The oint is that if, are under the critical hyerbola and s, t are chosen as in (3.13), then it is ossible to find (exlicitly) a function g (R N ) such that if u satisfies u(x) = g(x), x R N, then u does not belong to the fractional Sobolev sace H s (R N ), and resectively a function f +1 (R N ) such that the solution of v = f is not in H t. We recall that a function w is in H s (R N ) if and only if w 2 (R N ) and its Fourier transform ŵ(ξ) is such that ξ s ŵ(ξ) 2 (R N ). Then, assuming (4.22) does not hold we show we can erturb and cut off the functions f, g, to construct a seuence w = (f h, g h ) such that w = 1 and we can control the corresonding R h g h, S f in a way which yields a contradiction for small. Proof of emma 4.1. et us suose (4.22) does not hold, that is, there exists C 0 > 0 such that T w, w dx C 0, for each w E with w = 1. We start by giving some results from the theory of Fourier transforms, which we shall use. The next theorem is a standard fact from the theory of Fourier transforms of distributions. Theorem 2 Suose the function u 0 has slow growth, that is, there exists m N such that u 0 (x) dx <. (4.23) (1 + x ) m R N Then the Fourier transform û exists and belongs to the class of temered distributions S. In addition if φ C c (R N ) is such that φ 1 in B 1, φ 0 in R N \ B 2 and we set φ n (x) = φ(x/n) then φ n u û in S. We shall use the Fourier transform of the function w 0 (x) = x 2, 14

16 and its owers. It is a well-known fact from Fourier analysis that for any α > 0 we have ŵ0 α (ξ) = C(N, α) ξ α N 2 K N α( ξ ), (4.24) 2 (this is for instance formula (3.11) in [24]) ; here K ν (z) is the modified Bessel function of the second kind, given by K ν (z) = Γ(ν + 1) 2 z ν π 0 cos(t) (t 2 + z 2 ) ν+ 1 2 dt = Γ(ν ) π z ν 0 cos(sz) (1 + s 2 ) ν+ 1 2 Standard analysis shows that K ν (z) > 0, K ν (z) C (R\{0}), K decays exonentially as z, and, most imortantly, K ν (z) const. z ν as z 0. Hence ds. ŵ α 0 (ξ) C(N, α) ξ 2α N as ξ 0. (4.25) We now fix > and > such that, are still under the critical 1 hyerbola, > 1 2. We set N so that in articular α = N 2( + 1), β = N 2( + 1), et u 0, v 0 be the solutions of w α 0 (R N ), w β 0 +1 (R N ). u 0 = k 1 w α 0, v 0 = k 2 w β 0 in R N, (4.26) where k 1 = k 1 (,, N) := w α 0 1, k 2 = k 2 (,, N) := w β By standard PDE theory u 0 and v 0 are functions which belong to some ebesgue saces over R N (see for instance Theorem 10.2 (i) in [25]), which in articular imlies that they have slow growth, as in (4.23) (by the Hölder ineuality). Hence Theorem 2 alies, and, by taking the Fourier transform on both sides of the euations in (4.26) we get û 0 (ξ) = k 1 ξ 2 ŵ α 0 (ξ), v 0 (ξ) = k 2 ξ 2 ŵ β 0 (ξ). (4.27) Note that û 0, v 0 are ositive. emma 4.2 We have R N ξ 2 û 0 (ξ) v 0 (ξ) dξ =. 15

17 Proof. By (4.24) and (4.27) we have ξ 2 û 0 (ξ) v 0 (ξ) C(N, α, β) ξ 2(α+β 1 N) as ξ 0. However, by the choice of α and β that we made α + β 1 N = N ( ) N + 1 = N ( ) 1 N + 1 < N ( ) 1 N = N 2 N 2, and the lemma follows. We set u n = φ n u 0 Cc (R N ), v n = φ n v 0 Cc (R N ), where φ n is a function as in Theorem 2, and g n, := u n + b( x)u n, f n, := v n + b( x)v n. Since u n, v n have comact suort and b(0) = 0 for each fixed n we have Clearly g n, u n in, fn, v n in +1 as 0. u n u 0 in, vn v 0 in as n, and, recalling that we have taken u 0, v 0 so that u 0 = u 0 +1 = 1, we see that we can find n 0 such that for each n n 0 there exists n for which 2 g n, 1 2, 2 f n, , if < n. Now set g n, = g n,, f n, = 2 gn, f n, 2 fn, +1 and w n, = (f n,, g n, ). So w n, E and w n, X = 1. By the hyothesis we made T w n,, w n, C 0, for all n n 0 and all < n. 16,

18 On the other hand, setting k 1 n, = 2 g n, g n, (by the above k n, (1/8, 2)), we have T w n,, w n, = k n, ( f,n R g n, + g n, S fn, ) = k n, u n ( v n ) + v n ( u n ) + 2b( x)u n v n k n, û n( vn ) + v n( un ) = 2k n, ξ 2 φ n u 0 (ξ) φ n v 0 (ξ) dξ, where we used Parseval s identity and the ositivity of b, u n, v n. Hence ξ 2 φ n u 0 (ξ) φ n v 0 (ξ) dξ 4C 0. (4.28) Note that the definition of the Fourier transform imlies φ n u 0 (ξ) û 0 (ξ) for each ξ 0. Actually (see for instance Theorems 2.16, 5.3, 5.8 in [24]) φ n u 0 = φ n û 0 û 0 in any ebesgue sace to which belongs û 0, and similarly for v 0. Recall we have exlicit exressions for û 0, v 0 and know that they are strictly ositive, behave like ξ to a negative ower as ξ 0 and decay exonentially as ξ. It is then simle to check that the negative art of ξ 2 φ n u 0 (ξ) φ n v 0 (ξ) is bounded by an integrable function indeendently of n, so Fatou s lemma alies to (4.28) and gives a contradiction with emma 4.2. Alternatively, one can rove that Fatou s lemma alies to (4.2) by noticing that the integrand in this ineuality is (φ n u 0 ). (φ n v 0 ) and this scalar roduct is ositive, since φ n, u 0 and v 0 are ositive, radial, and decreasing functions. This comletes the roof of Theorem 1. 5 Aendix In this aendix we verify estimate (4.20), which we used in emma 3.4. First, we note that for w = (f, g) Ψ (tw) = A α tα + B β tβ C 2 t2, (5.29) where α = ( + 1)/, β = ( + 1)/, 17

19 A = g, B = f +1, C = T w, w. Denoting the right hand side of (5.29) by h(t), it is easy to check that max{ψ (tw), t 0} = h( t), for some t > 0 if and only if h ( t) = 0, that is t 2 = A C t α + B C t β. This imlies that there exists a ositive constant K such that [ (A ) 1 ( ) ] 1 2 α B 2 β t K +. C C Then, for some constant K > 0, h( t) = A( 1 α 1 2 ) t α + B( 1 β 1 2 ) t β [ K A 2 2 α C α 2 α [ = K A 2 2 α C α 2 α + AB α 2 β C α 2 β + A C α 2 C B α 2 β + BA β 2 α C β 2 α αβ 2(2 β) By using the Young ineuality, we obtain A C α 2 C B α 2 β αβ 2(2 β) + B C β 2 C A β 2 α αβ 2(2 α) ( + B C β 2 C A C α 2 ( + ] + B 2 2 β C β 2 β A β 2 α αβ 2(2 α) ) ( 2 2 α + B C β 2 ) 2 2 β ] + B 2 2 β. C β 2 β B 2 β α αβ C 2(2 β) ( + ) 2 α A 2 α β αβ C 2(2 α) ) 2 β. Thus, ( h( t) 2K A 2 α C ) α ( ) 2 α B 2 β [ ] 2 β β + 2K A 2 α + B 2 γ β, C C { where γ = max α, β 2 α 2 β }. 18

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