Periodic solutions for a pseudo-relativistic Schrödinger equation

Linear Algebra and its Applications 466 (2015) 102–116 Nonlinear Analysis 120 (2015) 262–284

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Linear Algebra and its Applications Nonlinear Analysis www.elsevier.com/locate/laa www.elsevier.com/locate/na

Inverse problem ofSchr¨ Jacobi matrix Periodic solutions for eigenvalue a pseudo-relativistic odinger equation with mixed data Vincenzo Ambrosio Dipartimento di Matematica e Applicazioni, Universit` a degli Studi “Federico II” di Napoli, via Cinthia, 80126 Napoli, Italy 1

Ying Wei

article

Department of Mathematics, Nanjing University of Aeronautics and Astronautics,

210016, PRaChina bstract i n fNanjing o

Article history: a r t i c l e iWe n fstudy o the existencea and b s the t rregularity a c t of non trivial T -periodic solutions to the Received 11 December 2014 Accepted 12 March 2015 following nonlinear pseudo-relativistic Schr¨ odinger equation Communicated by Enzo Mitidieri Article history: In this paper,the inverse eigenvalue problem of reconstructing   Received 16 January 2014 a Jacobi matrix from its eigenvalues, its leading principal Keywords: −∆x submatrix + m2 − mandu(x) = of f (x, u(x)) in (0, Tof )N its submatrix (0.1) Accepted 20 September 2014 part the eigenvalues Pseudo-relativistic Schr¨ odinger online 22 October 2014 Available is considered. The necessary and sufficient conditions for equation Submitted by Y. Weiwhere T > 0, m is a non negative real number, f is a regular function satisfying the existence and uniqueness of the solution are derived. Periodic solutions the Ambrosetti–Rabinowitz conditiona and a polynomial growth rate pnumerical for some Furthermore, numerical algorithm andat some Linking Theorem MSC: 1 < p < 2♯ − 1. Weexamples investigate such problem using critical point theory after are given. 15A18 transforming it to an elliptic equation in the infinite half-cylinder )N × (0,Inc. ∞) © 2014 Published(0, byTElsevier 15A57

Keywords: Jacobi matrix Eigenvalue Inverse problem Submatrix

with a nonlinear Neumann boundary condition. By passing to the limit as m → 0 in (0.1) we also prove the existence of a non trivial T -periodic weak solution to (0.1) with m = 0. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction In this paper we are concerned with periodic solutions for a nonlinear pseudo-relativistic Schr¨odinger equation. Particularly, we are looking for a function u satisfying the nonlinear problem  N       −∆x + m2 − m u(x) = f (x, u(x)) in (0, T )N = (0, T ) (1.1) i=1    N u(x + ei T ) = u(x) for all x ∈ R , i = 1, . . . , N where T > 0 is fixed, m ≥ 0 and (ei ) is the canonical basis in RN . √ 2 is defined [email protected]. The operator −∆E-mail as follows: let u ∈ CT∞ (RN ), that is u is infinitely differentiable in x + m address: 1 Tel.: +86 13914485239. RN and T -periodic in each variable. Then u has a Fourier series expansion: http://dx.doi.org/10.1016/j.laa.2014.09.031 0024-3795/© 2014 Published by Elsevier iωk·x Inc.  e u(x) = ck √ (x ∈ RN ) N T N k∈Z E-mail address: [email protected]. http://dx.doi.org/10.1016/j.na.2015.03.017 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

263

where ω=

2π T

and ck = √

1



TN

(0,T )N

u(x)e−iωk·x dx

(k ∈ ZN )

are the Fourier coefficients of u. √ The operator −∆x + m2 is defined by setting

For u =



k∈ZN

iωk·x

ck e√

TN

   eiωk·x −∆x + m2 u = ck ω 2 |k|2 + m2 √ . TN k∈ZN  iωk·x and v = k∈ZN dk e√ N , we have that T   Q(u, v) = ω 2 |k|2 + m2 ck dk

(1.2)

k∈ZN

can be extended by density to a quadratic form on the Hilbert space      eiωk·x 2 N 2 |k|2 + m2 |c |2 < ∞ . √ ω ∈ L (0, T ) : Hm = u = c k k T TN k∈ZN k∈ZN We assume that the nonlinear term f : RN × R → R in Eq. (1.1) satisfies the following conditions: (f1) f (x, t) is locally Lipschitz-continuous in RN × R; (f2) There exist a1 , a2 > 0 and p ∈ (1, 2♯ − 1) such that |f (x, t)| ≤ a1 + a2 |t|p

(f3) (f4) (f5) (f6)

∀t ∈ R ∀x ∈ RN .

Here the critical exponent 2♯ is infinite if N = 1 and N2N −1 if N ≥ 2; f (x,t) N lim|t|→0 |t| = 0 uniformly in x ∈ R ; There exist µ > 2 and r > 0 such that 0 < µF (x, t) ≤ tf (x, t) for all |t| ≥ r and for all x ∈ RN , where t F (x, t) = 0 f (x, s)ds; f is T -periodic in each variable xi , that is f (x + ei T, t) = f (x, t) for every x ∈ RN , t ∈ R and i = 1, . . . , N ; tf (x, t) ≥ 0 for any x ∈ RN and t ∈ R.

We remark that the hypothesis (f3) guarantees that (1.1) possesses the trivial solution u ≡ 0. The hypothesis (f4) gives information about the behavior of f (x, u) and F (x, u) at u = ∞. Indeed, a straightforward computation shows that, by (f4), there exist two constants a3 , a4 > 0 such that F (x, u) ≥ a3 |u|µ − a4

for x ∈ RN and t ∈ R.

(1.3)

Since µ > 2, (1.3) and (f2) imply that F (x, u) grows superquadratically and f (x, u) grows superlinearly as |u| → ∞. As a model for f we can take f (x, u) = g(x)|u|p−1 u, where g is a smooth positive T -periodic function. We observe that hypotheses (f1)–(f4) are standard when we deal with superlinear second order elliptic partial differential equations: see for example [21,26,27]. √ The non-local operator −∆x + m2 in RN plays an important role in relativistic quantum mechanics. Indeed the Hamiltonian for a (free) relativistic particle of momentum p and mass m is given by  H = c2 |p|2 + m2 c4 and with the usual quantization rule p → −i~∇, we get the so called pseudo-relativistic Hamiltonian operator and the associate free Schr¨ odinger equation  ∂ψ ˆ = − ~2 c2 ∆x + m2 c4 ψ. i = Hψ ∂t

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V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

Then choosing ~ = c = 1 we obtain the operator above mentioned. For a discussion of the main properties √ of the operator −∆x + m2 we refer to [17]. For physical models involving this operator one can see the works of Lieb and Yau [18,19] where in the first they study boson stars and in the second the stability of relativistic matter. More recently Fr¨ ohlich, Jonsson and Lenzmann [13] study the existence of solitary wave solutions of the pseudo-relativistic Hartree equation  i∂t ψ = ( −∆x + m2 − m)ψ − (|x|−1 ∗ |ψ|2 )ψ on R3 ; (see also [12,15,14] for related models). √ From a probabilistic point of view, the operator −( −∆x + m2 − m) is strictly connected with the potential theory: it is the infinitesimal generator of a Levy process, the so called the 1-stable relativistic process (see [11,22]). Recently the study of nonlinear equations involving a fractional Laplacian (−∆x )α has attracted the attention of many mathematicians, since it appears in many different contexts as phase transitions, optimization, finance, minimal surfaces and others. Caffarelli, Roquejoffre, Sire [8] and Caffarelli, Salsa, Silvestre [9] investigated free boundary problems of a fractional Laplacian. Silvestre [24] obtained some regularity results for the obstacle problem of the fractional Laplacian. Cabr´e and Sol´a Morales [6] studied an analogue of the De Giorgi conjecture for the equation (−∆x )α u = −G′ (u)

in RN

(1.4)

when α = 21 and G ∈ C 2 (R) has only two absolute minima. The same problem with α ∈ (0, 1) has been studied by Sire and Valdinoci in [25] and Cabr´e and Sire in [4,5]. Cabr´e and Tan [7] proved the existence of positive solutions to the problem  p   −∆x u = |u| in Ω (1.5) u = 0 on ∂Ω   u > 0 in Ω N +1 where Ω ⊂ RN is a smooth bounded domain and 1 ≤ p < N −1 (if N > 1), and Servadei and Valdinoci [23] dealt with the existence of non-trivial solutions of the following problem:  (−∆x )s u − λu = f (x, u) in Ω (1.6) u = 0 in RN \ Ω

where Ω ⊂ RN is a Lipschitz bounded domain, s ∈ (0, 1), λ is a real parameter and f (x, u) is a Carath´eodory function which behaves like u|u|p−2 for some 2 < p < N2N −2s . We can note that the above problems (1.5) and (1.6) can be seen as the fractional analogue of the classical problem  −∆x u − λu = f (x, u) in Ω (1.7) u = 0 on ∂Ω . This last problem has been also investigated with periodic boundary conditions by using variational and topological methods; see for instance [1,16,20,21,26] and references therein. The aim of the present paper is to study an analogue to (1.7) with periodic boundary conditions, when √ we replace −∆x by −∆x + m2 . The first result is the following: Theorem 1. Let m > 0. Let f : RN +1 → R be a function verifying the conditions (f1)–(f6). Then there exists at least a function um ∈ C 1,α (RN ) for some α ∈ (0, 1), T -periodic which satisfies the problem (1.1). One of the main difficulties of the analysis of the problem (1.1) is the nonlocal character of the involved operator. To circumvent this obstacle, we use the approach proposed by Caffarelli and Silvestre [10],

V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

265

which consists to realize the nonlocal problem (1.1) into a local problem in one more dimension via the Dirichlet–Neumann map. As explained in detail in Section 3, for u ∈ Hm T one can find a unique weak solution v ∈ Xm to the problem T   −∆v + m2 v = 0 in ST = (0, T )N × (0, ∞)  (1.8) v|{xi =0} = v|{xi =T } on ∂L ST = ∂(0, T )N × [0, +∞)   v(x, 0) = u(x) on ∂ 0 ST = (0, T )N × {0} where the boundary condition on ∂ 0 ST is in the sense of trace, and Xm T is defined as the completion of N +1 ∞ functions C (R+ ) and T -periodic in each xi , with respect to the norm  ∥v∥2Xm = |∇v|2 + m2 v 2 dxdy. T

ST

Note that in (1.8) the notation v|{xi =0} = v|{xi =T } on ∂L ST means v(x1 , . . . , xi−1 , 0, xi+1 , . . . , xN , y) = v(x1 , . . . , xi−1 , T, xi+1 , . . . , xN , y) for every i ∈ {1, . . . , N } and y ≥ 0. Furthermore, − lim+ y→0

 ∂v (x, y) = −∆x + m2 u(x) ∂y

∗ in (Hm T ) ,

in a weak sense. In order to find solutions of (1.1) and to prove their regularity, we will exploit this fact and look for solutions v ∈ Xm T to  −∆v + m2 v = 0 in ST    v|{xi =0} = v|{xi =T } on ∂L ST (1.9)  ∂v  0  = mv + f (x, v) on ∂ ST . ∂ν The variational structure of the problem (1.9) allows us to obtain the existence of T -periodic solutions vm through known variational methods, namely the Linking theorem. Such solutions are obtained as critical points in Xm T of the functional Jm associated to (1.9), that is    1 m Jm (v) = |∇v|2 + m2 v 2 dxdy − |v|2 dx − F (x, v) dx. 2 2 ∂ 0 ST ST ∂ 0 ST When m is sufficiently small, we are able to prove uniform estimates on critical levels αm of the functionals Jm . These estimates allows us to deduce uniform estimates on the solutions vm , and so we can pass to the limit as m → 0 in (1.9). As a consequence we can show the existence of a nontrivial solution to the problem  −∆x u(x) = f (x, u(x)) in (0, T )N (1.10) u(x + ei T ) = u(x) for all x ∈ RN , i = 1, . . . , N. This result can be stated as Theorem 2. Under the same assumptions of Theorem 1 we can find a non trivial T -periodic weak solution to the problem (1.10). The paper is organized as follows. In Section 2 we give some basic results concerning the fractional space and the nonlinear term f . In Section 3 we show that it is possible to transform the problem (1.1) in a Neumann elliptic problem. In Section 4 we prove the existence of weak solutions of the elliptic problem (1.9) through the Theory of Critical Point. In Section 5 we study the regularity of above critical points and we give the proof of Theorem 1. Finally, in the last section we find a nontrivial periodic solution to the problem (1.10). Hm T

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V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

2. Preliminaries In this section we collect preliminary facts for future reference. Firstly we denote the upper half-space in RN +1 by +1 RN = {(x, y) ∈ RN +1 : x ∈ RN , y > 0}. + +1 Let ST = (0, T )N ×(0, ∞) be the half-cylinder in RN with basis ∂ 0 ST = (0, T )N ×{0} and we denote by + N ∂L ST = ∂(0, T ) × [0, +∞) its lateral boundary. With ∥v∥Lr (ST ) we always denote the norm of v ∈ Lr (ST ) and with |u|Lr (0,T )N the Lr (0, T )N norm of u ∈ Lr (0, T )N . Denote by CT∞ (RN ) the space of functions u ∈ C ∞ (RN ) such that u is T -periodic in each variable, that is

for all x ∈ RN , i = 1, . . . , N.

u(x + ei T ) = u(x) Let u ∈ CT∞ (RN ). Then we know that u(x) =

eiωk·x ck √ TN k∈ZN 

for all x ∈ RN ,

where 2π ω= T

and ck = √



1 TN

(0,T )N

u(x)e−iωk·x dx

(k ∈ ZN )

N ∞ are the Fourier coefficients of u. We define the fractional Sobolev space Hm T as the closure of CT (R ) under the norm   |u|2Hm := ω 2 |k|2 + m2 |ck |2 . (2.1) T

k∈ZN

When m = 1, we set HT = H1T and ∥ · ∥HT = ∥ · ∥H1T . Now we introduce the functional space Xm T defined as the completion of   +1 +1 +1 CT∞ (RN ) = v ∈ C ∞ (RN ) : v(x + ei T, y) = v(x, y) for every (x, y) ∈ RN , i = 1, . . . , N + + + under the norm ∥v∥2Xm =



T

|∇v|2 + m2 v 2 dxdy.

(2.2)

ST

If m = 1, we set XT = X1T and ∥ · ∥XT = ∥ · ∥X1T . We begin proving that it is possible to define a trace m operator from the space Xm T to the fractional space HT : m Theorem 3. There exists a bounded linear operator Tr : Xm T → HT such that: +1 (i) Tr(v) = v|∂ 0 ST for all v ∈ CT∞ (RN ) ∩ Xm + T ; m m for every v ∈ X (ii) |Tr(v)|Hm ≤ ∥v∥ ; X T T T (iii) Tr is surjective. +1 Proof. Let v ∈ CT∞ (RN ) ∩ Xm + T . Then using the Fourier series we can write

v(x, y) = where ck (y) =

√1 TN



(0,T )N

eiωk·x ck (y) √ TN k∈ZN

v(x, y)e−iωk·x dx.



N +1

in R+

,

V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

267

Note that ck ∈ C ∞ ([0, ∞)) ∩ H 1 ((0, ∞)), therefore ck (y) → 0 as y → ∞, for all k ∈ ZN . By the Fundamental Theorem of Calculus and using the H¨older inequality we have  ∞ d 2 |ck (0)| = − |ck (y)|2 dy dy 0  ∞ = −2 ck (y)c′k (y)dy 0  21  ∞  12  ∞ |c′k (y)|2 dy . ≤2 |ck (y)|2 dy 0

0



ω 2 |k|2

m2 ,

using the Young inequality and summing over ZN , we obtain    ∞   (2.3) ω 2 |k|2 + m2 |ck (y)|2 + |c′k (y)|2 dy. ω 2 |k|2 + m2 |ck (0)|2 ≤

Multiplying both members by

+

0

k∈ZN

k∈ZN

Thanks to Parseval Identity, we have  

∥v∥2L2 (ST ) =

k∈ZN

∥∇x v∥2L2 (ST )

∞

 

=

k∈ZN

∥vy ∥2L2 (ST ) =

|ck (y)|2 dy,

0

  k∈ZN

0

0 ∞

∞

(2.4)

ω 2 |k|2 |ck (y)|2 dy,

(2.5)

|c′k (y)|2 dy.

(2.6)

Taking into account (2.3)–(2.6) we obtain ∥v∥2Xm = ∥∇x v∥2L2 (ST ) + ∥vy ∥2L2 (ST ) + m2 ∥v∥2L2 (ST ) T     ∞  = ω 2 |k|2 + m2 |ck (y)|2 + |c′k (y)|2 dy k∈ZN

≥

0

 

ω 2 |k|2 + m2 |ck (0)|2 .

(2.7)

k∈ZN

By density we deduce that (2.7) is verified for each v ∈ Xm T . This allows us to say that is well defined a bounded linear operator m Tr : Xm T → HT . m Finally we prove that Tr(Xm T ) = HT , that is Tr is surjective.  iωk·x Let u(x) = k∈ZN ck e√ N ∈ Hm T where ck are the Fourier coefficients of u. Consider the function T

v(x, y) =

eiωk·x −√ω2 |k|2 +m2 y ck √ e TN k∈ZN 

(2.8)

which is clearly smooth for y > 0. We want to show that v ∈ Xm T and Tr(v) = u. Observe that v is T -periodic in each variable xi and v solves the equation −∆v + m2 v = 0 in ST . Moreover v(x, y) → u(x) as y → 0+ in L2 (0, T )N . In fact fixed ε > 0, there exist δε > 0 and kε ∈ N such that √ 2 2 2   ε ε |ck |2 < and |ck |2 (1 − e− ω |k| +m y ) < 2 2 |k|>kε

|k|≤kε

for 0 < y < δε .

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V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

Thus for every 0 < y < δε  √ 2 2 2 √ 2 2 2   |ck |2 (1 − e− ω |k| +m y )2 |v(x, y) − u(x)|2 dx = |ck |2 (1 − e− ω |k| +m y )2 + (0,T )N

|k|>kε

|k|≤kε

 ε < + |ck |2 < ε. 2 |k|>kε

Let us check that v ∈ Xm T . By Parseval Identity we can see that ∥v∥2Xm = ∥∇x v∥2L2 (ST ) + ∥vy ∥2L2 (ST ) + m2 ∥v∥2L2 (ST ) T   ∞ √  2 2 2 2 2 2 2 e−2 ω k +m y dy |ck | ω |k| + m =2 0

k∈ZN

=

 

ω 2 |k|2 + m2 |ck |2 < ∞.

(2.9)

k∈ZN m This proves that Hm T ⊆ Tr(XT ).



Then we have the following compact embedding: Theorem 4. Let 1 ≤ q < 2♯ for N ≥ 2 and 1 ≤ q < ∞ for N = 1. Then Tr(Xm T ) is compactly embedded in Lq (0, T )N . m Proof. By Theorem 3 we know that Tr(Xm T ) is embedded with continuity in HT . To conclude the proof, it q N m is enough to prove that HT is compactly embedded in L (0, T ) .  iωk·x 2N 1 1 ′ Let u = k∈ZN ck e√ N ∈ Hm T . Fix N +1 < r < 2 and let r be its conjugate exponent, that is r + r ′ = 1. T By the H¨ older inequality we can see that    r1  r1  r  r |ck |r ( ω 2 |k|2 + m2 ) 2 ( ω 2 |k|2 + m2 )− 2 = |ck |r k∈ZN

k∈ZN

≤ |u|Hm T

   2−r r 2r ( ω 2 |k|2 + m2 )− 2−r

(2.10)

k∈ZN

and the last series is finite since r > ′ u ∈ Lr (0, T )N and

2N N +1 .

By Theorem of Hausdorff and Young [28] we deduce that

 1  r2 −1    r1 |u|Lr′ (0,T )N ≤ √ |ck |r . TN k∈ZN

(2.11)

Hence, by (2.10) and (2.11), we have |u|Lq (0,T )N ≤ C



|ck |2



ω 2 |k|2 + m2

 12

(2.12)

k∈ZN

holds for every 2 ≤ q < 2♯ . Using (2.12) and interpolation inequality we obtain that for all q ∈ [2, 2♯ ) |u|Lq (0,T )N ≤ C|u|θL2 (0,T )N

 k∈ZN

for some real positive number θ ∈ (0, 1).

|ck |2

1−θ  ω 2 |k|2 + m2 ,

(2.13)

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V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

2 N At this point we prove that Hm T b L (0, T ) , for any N ∈ N.

Let uj ⇀ 0 in Hm T . Then lim |cjk |2

j→∞

 ω 2 |k|2 + m2 = 0 ∀k ∈ ZN

(2.14)

and 

|cjk |2

 ω 2 |k|2 + m2 ≤ C

∀j ∈ N.

(2.15)

k∈ZN 1

Fix ε > 0. Then there exists ν > 0 such that (ω 2 |k|2 + m2 )− 2 < ε for |k| > ν. By (2.15) we have  j  j  j |ck |2 = |ck |2 + |ck |2 k∈ZN

|k|≤ν

=

 |k|≤ν

≤



|k|>ν

|cjk |2

+



1

1

|cjk |2 (ω 2 |k|2 + m2 ) 2 (ω 2 |k|2 + m2 )− 2

|k|>ν

|cjk |2 + Cε.

|k|≤ν

By (2.14) we deduce that



|k|≤ν

|cjk |2 < ε for j large. So uj → 0 in L2 (0, T )N .

q N 2 N and (2.13) we can conclude that Hm Then using Hm T is compactly embedded in L (0, T ) T b L (0, T ) ♯ for every q ∈ [2, 2 ). 

q N for any q ≤ 2♯ Remark 1. It is possible to prove the existence of a continuous embedding Hm T ⊂ L (0, T ) (see for instance [2]).

Finally, we give some elementary results which will be used in the sequel. We use the growth conditions (f2), (f3) and (f4) to deduce some bounds from above and below for the nonlinear term and its primitive. This part is quite standard and the proofs of the following lemmas can be found in [21] (see also [23]). Lemma 1. Let f : [0, T ]N × R → R satisfying conditions (f1)–(f3). Then, for any ε > 0 there exists Cε > 0 such that |f (x, t)| ≤ 2ε|t| + (p + 1)Cε |t|p

∀t ∈ R ∀x ∈ [0, T ]N

(2.16)

and |F (x, t)| ≤ ε|t|2 + Cε |t|p+1

∀t ∈ R ∀x ∈ [0, T ]N .

(2.17)

Lemma 2. Assume that f : [0, T ]N × R → R satisfies conditions (f1)–(f4). Then, there exist two constants a3 > 0 and a4 > 0 such that F (x, t) ≥ a3 |t|µ − a4

∀t ∈ R ∀x ∈ [0, T ]N .

(2.18)

3. Problem in the cylinder In this section we will show that it is possible to transform the problem (1.1) in an elliptic problem with Neumann condition on the boundary (0, T )N × {0} and periodic conditions on the lateral boundary of the cylinder.

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V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

More precisely we prove the following m Theorem 5. Let u ∈ Hm T . Then there exists a unique v ∈ XT solution to the problem  2   −∆v + m v = 0 in ST v|{xi =0} = v|{xi =T } on ∂L ST   v(·, 0) = u on ∂ 0 ST

(3.1)

where the boundary condition on ∂ 0 ST is in the sense of trace. In addition − lim+ vy (x, y) = y→0

Proof. Let u(x) =

 ∗ −∆x + m2 u(x) in (Hm T ) .

(3.2)

iωk·x

ck e√ N ∈ CT∞ (RN ). Consider the following minimizing problem: T    |∇v|2 + m2 v 2 dxdy : v ∈ Xm inf and Tr(v) = u . T



k∈ZN

(3.3)

ST m By lower weak semi-continuity of the Xm T -norm and by Theorem 4 we can find a minimizer v ∈ XT . Moreover from the strict convexity of the functional in (3.3), we can see that this minimizer is unique. It follows that v is a weak solution to  2   −∆v + m v = 0 in ST (3.4) v|{xi =0} = v|{xi =T } on ∂L ST   v(·, 0) = u on ∂ 0 ST .  By standard elliptic regularity, we deduce that v is smooth for y ≥ 0. We may write v(x, y) = k∈ZN  −iωk·x iωk·x ck (y) e√ N , where ck (y) = (0,T )N v(x, y) e √ N dx. Since u is the trace of v, ck (0) are the Fourier coefficients T T of u. Moreover, for any k ∈ ZN , ck (y) satisfies the equation

− c′′k (y) + (ω 2 |k|2 + m2 )ck (y) = 0

for y > 0.

(3.5)

Therefore we deduce that v(x, y) =

eiωk·x −√ω2 |k|2 +m2 y . e ck √ TN k∈ZN 

(3.6)

Hence    ∂v  eiωk·x 2 |k|2 + m2 √ ω := − lim v (x, y) = c y k y→0+ ∂ν ∂ 0 ST TN k∈ZN  = −∆x + m2 u

(3.7) (3.8)

and similarly as in (2.9), we obtain ∥v∥2Xm =

 

T

ω 2 |k|2 + m2 |ck |2 = |u|2Hm . T

(3.9)

k∈ZN

By density we get the desired result.



We will call v the extension of u, and we will denote it with extm (u). Hence to study the problem (1.1) it is equivalent to study the solutions v ∈ Xm T to the problem  2 −∆v + m v = 0 in ST    v|{xi =0} = v|{xi =T } on ∂L ST    ∂v = mv + f (x, v) on ∂ 0 ST . ∂ν

(3.10)

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271

More precisely, we will say that u ∈ Hm T is a weak solution to (1.1) if and only if its extension v = extm (u) ∈ m XT is a weak solution to (3.10), that is if   ∇v∇η + m2 vη dxdy = [mv(x, 0) + f (x, v(x, 0))]η(x, 0) dx (3.11) (0,T )N

ST

holds for all η ∈ Xm T . Remark 2. In order to simplify notation in what follows we will denote w for the function defined in the cylinder ST as well as for its trace Tr(w) on ∂ 0 ST . 4. Linking solutions In this section we prove the existence of weak solutions to the problem (3.10) by using the Linking Theorem due to Rabinowitz [21]: Theorem 6. Let (X, ∥ · ∥) be a real Banach space with X = Y J ∈ C 1 (X, R) be a functional satisfying the following conditions:



Z, where Y is finite dimensional. Let

• J satisfies the Palais–Smale condition, • There exist η, ρ > 0 such that inf{J(v) : v ∈ Z and ∥v∥ = η} ≥ ρ, • There exist z ∈ ∂B1 ∩ Z, R > ρ and R′ > 0 such that J ≤0

on ∂A

where A = {v = y + sz : y ∈ Y, ∥y∥ ≤ R′ and 0 ≤ s ≤ R} and ∂A = {v = y + sz : y ∈ Y, ∥y∥ = R′ or s ∈ {0, R}}. Then J possesses a critical value c ≥ ρ which can be characterized as c = inf max J(γ(v)) γ∈Γ v∈A

where Γ = {γ ∈ C(A, X) : γ = Id on ∂A}. Let us consider the following functional    m 1 2 2 2 2 |∇v| + m v dxdy − |v| dx − F (x, v) dx Jm (v) = 2 2 ∂ 0 ST ST ∂ 0 ST m 1 m for v ∈ Xm T . First of all we can observe that Jm is well defined on XT and Jm ∈ C (XT , R) by Theorem 3 and by assumptions on f . Moreover, using Theorem 3, we deduce that the quadratic part of Jm is non-negative,

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that is for every v ∈ Xm T  m

(0,T )N



|v(x, 0)|2 dx ≤

|∇v|2 + m2 v 2 dxdy.

(4.1)

ST

Indeed this inequality is satisfied by each function in H 1 of the cylinder. In fact for every function v ∈ C 1 such that v(x, y) → 0 as y → +∞ and for every q ∈ [2, 2♯ ] we have    0 ∂ |v(x, y)|q dy |v(x, 0)|q dx = dx ∂y N N +∞ (0,T ) (0,T )  ∂    ≤q |v(x, y)|q−1  v(x, y)dxdy ∂y ST 2    12    ∂  12   ≤q |v(x, y)|2(q−1) dxdy (4.2)  v(x, y) dxdy ST ST ∂y where in the last inequality we have exploited the Cauchy–Schwarz inequality. Then taking q = 2 and using 2ab ≤ a2 + b2 for all a, b ≥ 0, we deduce that  m|v(·, 0)|2L2 (0,T )N ≤ 2m |v(x, y)| |∂y v(x, y)| dxdy ST

≤ 2m∥v∥L2 (ST ) ∥∂y v∥L2 (ST )

(4.3)

≤ ∥∂y v∥2L2 (ST ) + m2 ∥v∥2L2 (ST )

(4.4)

≤ ∥∇v∥2L2 (ST ) + m2 ∥v∥2L2 (ST ) .

(4.5)

By density we obtain that (4.1) holds for every function in H 1 (ST ). We can observe that ∥v∥2Xm − m|v(·, 0)|2L2 (0,T )N = 0

if and only if v(x, y) = ce−my a.e.

T

(4.6)

for some constant c ∈ R. In fact, taking the equality in (4.3)–(4.5) we deduce that m2 v 2 = Cvy2 , m∥v∥L2 (ST ) = ∥vy ∥L2 (ST ) and v(x, y) = h(y) for some h ∈ H 1 (0, ∞). These conditions force to be v(x, y) = ce−my for some c ∈ R. We note that Xm T admits the following decomposition  m Xm Zm T = YT T −my ⟩, dim Ym where Ym T = ⟨e T < ∞ and

  m Zm = v ∈ X : T T

(0,T )N

 v(x, 0) dx = 0 .

Now we give some lemmas to prove Linking hypothesis: Lemma 3. The functional Jm is nonpositive on Ym T . Proof. In view of (4.6) we have that ∥v∥2Xm − m|v(·, 0)|2L2 (0,T )N = 0 if v ∈ Ym T . Moreover F (x, 0) = 0 and T

d F (x, t) = tf (x, t) ≥ 0 for all t ∈ R. Therefore F (x, t) is increasing with by hypothesis (f6) we know that t dt respect to t ≥ 0 and decreasing with respect to t ≤ 0, that is F (x, t) ≥ 0 for every x ∈ [0, T ]N , t ∈ R. Hence Jm is nonpositive on Ym  T .

Lemma 4. There exist ηm , ρm > 0 such that inf{Jm (v) : v ∈ Zm = ηm } ≥ ρm > 0. T and ∥v∥Xm T

(4.7)

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273

Proof. Firstly, we prove that there exists a constant C > 0 (depending eventually on m and T ) such that ∥v∥2e := ∥v∥2Xm − m|v(·, 0)|2L2 (0,T )N ≥ C∥v∥2Xm T

T

(4.8)

m m for every v ∈ Zm T (that is ∥ · ∥e is equivalent to ∥ · ∥XT on ZT ).

By contradiction suppose that there exists (vn ) ⊂ Zm T such that n∥vn ∥2e < ∥vn ∥2Xm

for all n ∈ N.

T

(4.9)

m m Assuming that ∥vn ∥Xm = 1 we deduce that vn converges weakly in Xm T to some function v ∈ ZT (ZT is T weakly closed). Then using (4.9) we obtain

0 ≤ ∥vn ∥e ≤

1 →0 n

as n → +∞

and so 1 |vn (·, 0)|L2 (0,T )N → √ m

as n → +∞.

Therefore 0 ≤ ∥v∥e ≤ lim inf n→∞ ∥vn ∥e = 0, that is v = ce−my . But v ∈ Zm T , hence c = 0 and this gives a contradiction because of |v(·, 0)|L2 (0,T )N = √1m > 0. Now, we are able to prove (4.7). In fact, by Theorem 4, (4.1), (4.8) and (2.17) we can see that for all v ∈ Zm T    1 m Jm (v) = |∇v|2 + m2 v 2 dxdy − |v|2 dx − F (x, v) dx 2 2 ∂ 0 ST ST ∂ 0 ST C ≥ ∥v∥2Xm − ε|v(·, 0)|2L2 (0,T )N − Cε |v(·, 0)|p+1 Lp+1 (0,T )N T 2 C ε . ≥ ∥v∥2Xm − ∥v∥2Xm − cε ∥v∥p+1 Xm T T T 2 m Then, choosing ε ∈ (0, mC 2 ), we can find ηm , ρm > 0 such that inf{Jm (v) : v ∈ Zm = ηm } ≥ ρm > 0.  T and ∥v∥Xm T ′ Lemma 5. There exist z ∈ ∂B1 ∩ Zm T and Rm > ηm , Rm > 0 such that

max Jm = 0 and m

sup Jm < ∞,

∂AT

Am T

where m ′ Am ≤ Rm and 0 ≤ s ≤ Rm } T := {v = y + sz : y ∈ YT , ∥y∥Xm T

and m ′ ∂Am = Rm or s ∈ {0, Rm }}. T = {v = y + sz ∈ AT : ∥y∥Xm T

Proof. By Lemma 3 we know that Jm is nonpositive on Ym T . Let w(x, y) =

N 

sin(ωxi )e−my

i=1

and z =

w . ∥w∥Xm T

m We note that z ∈ Zm T , ∥z∥XT = 1 and

∥z∥2Xm − m|z(·, 0)|2L2 (0,T )N = T

2N π 2 =: C > 0. 2N π 2 + m2 T 2

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Then for every v ∈ Ym T



Rz

|v(·, 0)|µLµ (0,T )N ≥ C ′ =C

′

 (0,T )N



 µ2 |c + sz(x, 0)|2 dx 

2

(0,T )N

c dx +

= C ′ (T N c2 + C ′′ s2 )

(0,T )N

 µ2 s2 |z(x, 0)|2 dx

µ 2

≥ C(c2 + s2 ) 2 . µ

(4.10)

Therefore, if v = ce−my + sz ∈ YT Rz, using (2.18) and (4.10) we have       s2 m Jm (v) = |∇z|2 + m2 z 2 dxdy − |z|2 dx − F (x, v) dx 2 2 ∂ 0 ST ST ∂ 0 ST  m

≤C ≤

s2 − (a3 |v(·, 0)|µLµ (0,T )N − a4 T N ) 2

µ C 2 s + a4 T N − a3 C(s2 + c2 ) 2 . 2

(4.11)

Hence C 2 s + a4 T N − a3 Csµ , 2 and so we can find Rm > ηm such that Jm (y + sz) ≤ 0 for every s ≥ Rm and y√∈ Ym T . Let 0 ≤ s ≤ Rm . ′ N |c| ≥ R′ . Finally > 0 such that Jm (y + sz) ≤ 0 for every ∥y∥Xm By (4.11), there exists Rm = mT m T 2 Jm (v) ≤ CRm + a4 T N for v ∈ Am  T . Jm (y + sz) ≤

To obtain the existence of a critical value of Jm we must prove the Palais–Smale condition, that is: Lemma 6. Let c ∈ R and let (vn ) ⊂ Xm T be a sequence such that Jm (vn ) → c and

′ Jm (vn ) → 0

as n → ∞.

(4.12)

Then (vn ) has a strongly convergent subsequence in Xm T . Proof. We begin proving that (vn ) is bounded in Xm T . Fix β ∈ ( µ1 , 21 ). By Lemma 1 applied with ε = 1 we have     βf (x, vn )vn − F (x, vn )dx ≤ ((2β + 1)r2 + C1 (p + 2)rp+1 )T N =: κ0  ∂ 0 ST ∩{|vn |≤r}

(4.13)

and   

∂ 0 ST ∩{|vn |≤r}

  F (x, vn )dx ≤ (r2 + C1 rp+1 )T N = κ′0 .

(4.14)

Then, using (4.1), (4.12)–(4.14) and (2.18) we have, for n sufficiently large ′ c + 1 + ∥vn ∥Xm ≥ Jm (vn ) − β⟨Jm (vn ), vn ⟩ T   1 − β [∥vn ∥2Xm − m|vn (·, 0)|2L2 (0,T )N ] + [βf (x, vn )vn − F (x, vn )]dx = T 2 ∂ 0 ST  ≥ [βf (x, vn )vn − F (x, vn )]dx ∂ 0 ST   = [βf (x, vn )vn − F (x, vn )]dx + [βf (x, vn )vn − F (x, vn )]dx ∂ 0 ST ∩{|vn |≥r}

∂ 0 ST ∩{|vn |≤r}

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 ≥ (µβ − 1)

∂ 0 ST ∩{|vn |≥r}

 ≥ (µβ − 1)

∂ 0 ST

F (x, vn )dx − κ0

F (x, vn )dx − (µβ − 1)κ′0 − κ0

 = (µβ − 1)

∂ 0 ST

F (x, vn )dx − κ

(4.15)

≥ (µβ − 1)[a3 |vn (·, 0)|µLµ (0,T )N − a4 T N ] − κ   µ−2 ≥ (µβ − 1) a3 |vn (·, 0)|µL2 (0,T )N T −N 2 − a4 T N − κ.

(4.16)

where in the last inequality we have used the H¨ older inequality. Hence, exploiting (4.15) and (4.16) ∥vn ∥2Xm = 2Jm (vn ) + m|vn (·, 0)|2L2 (0,T )N + 2



T

2

≤ C1 + C2 (1 + C3 + ∥vn ∥Xm )µ + T

∂ 0 ST

F (x, vn )dx

2 (1 + c + ∥vn ∥Xm ) T µβ − 1

≤ C4 + C5 ∥vn ∥Xm T and so (vn ) is bounded in Xm T . Going if necessary to a subsequence, we can assume that vn converges weakly to some function v ∈ Xm . T Moreover, by Theorem 4, up to a subsequence, we have in Lp+1 (0, T )N

vn (·, 0) → v(·, 0)

(4.17)

N

vn (·, 0) → v(·, 0)

a.e. in (0, T )

|vn (x, 0)| ≤ h(x)

a.e. in (0, T )

N

(4.18) p+1

for n ∈ N, for some h ∈ L

N

(0, T ) .

(4.19)

By hypotheses (f1) and (f2), by (4.17)–(4.19) and taking into account the Dominated Convergence Theorem we get as n → ∞   f (x, vn )vn dx → f (x, v)v dx (4.20) ∂ 0 ST

∂ 0 ST

and 



∂ 0 ST

f (x, vn )v dx →

∂ 0 ST

f (x, v)v dx.

(4.21)

′ 2 N Exploiting the facts Jm (vn ) → 0, vn ⇀ v in Xm T , vn (·, 0) → v(·, 0) in L (0, T ) , (4.20) and (4.21) we can see that as n → ∞  ′ Jm (vn )vn → 0 ⇒ ∥vn ∥2Xm − m|vn (·, 0)|2L2 (0,T )N → f (x, v)v dx ∂ 0 ST

T

and ′ Jm (vn )v → 0 ⇒ ∥v∥2Xm − m|v(·, 0)|2L2 (0,T )N = T

 ∂ 0 ST

f (x, v)v dx.

So ∥vn ∥Xm → ∥v∥Xm as n → ∞. Finally using ∥vn ∥Xm → ∥v∥Xm and vn ⇀ v in Xm T we obtain T T T T ∥vn − v∥2Xm = ∥vn ∥2Xm + ∥v∥2Xm − 2⟨vn , v⟩Xm → 0.  T T

T

T

Taking into account Lemmas 3–6 we can apply Theorem 6, so we deduce that: Theorem 7. There exists at least one weak solution vm ∈ Xm T to the problem (3.10).

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5. Regularity of solutions This section is devoted to study the regularity of weak solutions to the problem (3.10). We begin proving the following result: Lemma 7. Let v ∈ Xm T be a weak solution to  −∆v + m2 v = 0 in ST    v|{xi =0} = v|{xi =T } on ∂L ST    ∂v = mv + f (x, v) on ∂ 0 ST . ∂ν Then v(·, 0) ∈ Lq (0, T )N for all q < ∞.

(5.1)

Proof. If N = 1 the thesis follows by Theorem 4. Let N ≥ 2. Now we proceed as in the proof of Theorem 3.2 in [12]. Since v is a critical point for Jm , we know that   ∇v∇η + m2 vη dxdy = mvη + f (x, v)η dx (5.2) ∂ 0 ST

ST

2β m for all η ∈ Xm T . Let w = vvK where vK = min{|v|, K}, K > 1 and β ≥ 0. Then w ∈ XT and taking η = w in (5.2) we deduce that   2β 2β |∇v|2 dxdy (|∇v|2 + m2 v 2 ) dxdy + 2βvK vK ST

DK,T

 =m

∂ 0 ST

2β v 2 vK dx +

 ∂ 0 ST

2β f (x, v)vvK dx

(5.3)

where DK,T = {(x, y) ∈ ST : |v(x, y)| ≤ K}. By direct computation we can see   β 2 |∇(vvK )| dxdy =

ST

ST

2β vK |∇v|2 dxdy +

 DK,T

2β (2β + β 2 )vK |∇v|2 dxdy.

(5.4)

Combining (5.3) and (5.4) we find that  β 2 β 2 2β ∥vvK ∥Xm = |∇(vvK )| + m2 v 2 vK dxdy T ST   2β 2 2 2 = vK [|∇v| + m v ]dxdy + S

≤ cβ

T  

ST

 = cβ

∂ 0 ST

 β  2β 2β 1 + v |∇v|2 dxdy 2 K DK,T   2β 2β vK [|∇v|2 + m2 v 2 ]dxdy + 2βvK |∇v|2 dxdy DK,T

2β 2β mv 2 vK + f (x, v)vvK dx

(5.5)

where cβ = 1 + β2 . Using Lemma 1 with ε = 1 we get 2β 2β 2β 2β mv 2 vK + f (x, v)vvK ≤ (m + 2)v 2 vK + (p + 1)C1 |v|p−1 v 2 vK

on ∂ 0 ST .

We also have that |v|p−1 = χ{|v|≤1} |v|p−1 + χ{|v|>1} |v|p−1 ≤ 1 + h on ∂ 0 ST where h ∈ LN (0, T )N . In fact if (p − 1)N < 2 then   χ{|v|>1} |v|N (p−1) dx ≤ ∂ 0 ST

∂ 0 ST

χ{|v|>1} |v|2 dx < ∞

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while if 2 ≤ (p − 1)N we have that (p − 1)N ∈ [2, N2N −1 ]. Therefore we have proved that there exist a constant N c = m + 2 + (p + 1)C1 and a function h ∈ L (0, T )N , h ≥ 0 and independent of K and β such that 2β 2β 2β mv 2 vK + f (x, v)vvK ≤ (c + h)v 2 vK

on ∂ 0 ST .

(5.6)

As a consequence of (5.5) and (5.6) we have β 2 ∥vvK ∥Xm T

 ≤ cβ

∂ 0 ST

2β (c + h)v 2 vK dx,

and taking the limit as K → ∞ (vK is increasing with respect to K) we get   β+1 2 2(β+1) ∥ |v| ∥Xm ≤ ccβ |v| dx + cβ h|v|2(β+1) dx.

(5.7)

For any M > 0, let A1 = {h ≤ M } and A2 = {h > M }. Then  h|v|2(β+1) dx ≤ M | |v(·, 0)|β+1 |2L2 (0,T )N + ε(M )| |v(·, 0)|β+1 |2L2♯ (0,T )N

(5.8)

∂ 0 ST

T

∂ 0 ST

∂ 0 ST

where ε(M ) =



hN dx A2

 N1

→ 0 as M → ∞. Taking into account (5.7), (5.8), we have

∥ |v|β+1 ∥2Xm ≤ cβ (c + M )| |v(·, 0)|β+1 |2L2 (0,T )N + cβ ε(M )| |v(·, 0)|β+1 |2L2♯ (0,T )N . T

(5.9)

Choosing M large so that ε(M )cβ C22♯ < 12 , using Theorem 3, Remark 1 and (5.9) we obtain | |v(·, 0)|β+1 |2L2♯ (0,T )N ≤ C22♯ ∥ |v|β+1 ∥2Xm ≤ 2C22♯ cβ (c + M )| |v(·, 0)|β+1 |2L2 (0,T )N . T

2N

Then we can start a bootstrap argument: since v(·, 0) ∈ L N −1 we can apply (5.10) with β1 + 1 = deduce that v(·, 0) ∈ L v(·, 0) ∈ L

2N k (N −1)k

(β1 +1)2N N −1

(0, T )

N

=L

2N 2 (N −1)2

(5.10) N N −1

to

N

(0, T ) . Applying (5.10) again, after k iterations, we find

(0, T )N , and so v(·, 0) ∈ Lq (0, T )N for all q ∈ [2, ∞).



Now we are ready to show that the weak solutions of (3.10) are H¨older continuous together with their partial derivatives up to the boundary of the cylinder (hence in the whole of the upper half-space). Theorem 8. Let v ∈ Xm T be a weak solution to the problem  2   −∆v + m v = 0 in ST  v|{xi =0} = v|{xi =T } on ∂L ST  ∂v   = g(x, v) on ∂ 0 ST ∂ν

(5.11)

+1 where g(x, v(x, 0)) = mv(x, 0) + f (x, v(x, 0)). Assume that v is extended by periodicity to the whole RN . + N +1

Then v ∈ C 1,α (R+

+1 ) ∩ C ∞ (RN ) for some α ∈ (0, 1). +

Proof. We proceed with a useful method introduced by Cabr´e and Sol´a Morales in [6], which consists of using the auxiliary function  y w(x, y) = v(x, s)ds. (5.12) 0

Then w weakly solves  2   −∆w + m w = g(x, v) in ST w|{xi =0} = w|{xi =T } on ∂L ST   w = 0 on ∂ 0 ST .

(5.13)

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Denoting again with w the T -periodic extension of w with respect to x to the whole RN , we can prove that w satisfies   2 ∇w∇η + m wη dxdy = g(x, v)η dxdy (5.14) +1 RN +

+1 RN +

+1 for all η ∈ Cc∞ (RN ), that is w is a weak solution to the Dirichlet problem +  +1 −∆w + m2 w = g in RN +

w=0

+1 on ∂RN . +

(5.15)

+1 Let η ∈ Cc∞ (RN ) and τ = (T, . . . , T ). Then we can see that +   ∞ ∇w(x, y)∇η(x, y) + m2 w(x, y)η(x, y)dxdy 0

RN

 

=

0

Tk

k∈ZN

∞



∇w(x, y)∇η(x, y) + m2 w(x, y)η(x, y)dxdy

(5.16)

where Tk = (0, T )N + kτ and kτ is a shortcut for (k 1 T, . . . , k N T ). Note that in this sum only a finite number of terms are not equal to zero since η is assumed to have a compact support. Making the change of variable x + kτ → x in Tk and using the T -periodicity of w and g, we obtain    ∞ ∇w(x, y)∇η(x, y) + m2 w(x, y)η(x, y) dxdy =

0

Tk

k∈ZN

 



(0,T )N

k∈ZN

 =

(0,T )N

 = =

0

(0,T )N

(0,T )N



k∈ZN

Tk



∞

RN

∇w(x, y)∇ψ(x, y) + m2 w(x, y)ψ(x, y) dxdy



   0

∇w(x, y)∇η(x + kτ, y) + m2 w(x, y)η(x, y) dxdy

g(x, v(x, 0))ψ(x, y) dxdy

0

 

=

0

∞



k∈ZN

=

∞



∞

0

0

∞

g(x, v(x, 0))η(x + kτ, y) dxdy

∞

g(x − kτ, v(x − kτ, 0))η(x, y) dxdy

g(x, v(x, 0))η(x, y) dxdy

(5.17)

 where we have set ψ(x, y) = k∈ZN η(x + kτ, y). Clearly this function is admissible since it is indefinitely differentiable, T -periodic in x and it vanishes near y = 0. Therefore taking into account (5.16) and (5.17) we deduce (5.14). Denote with wodd and godd the extension of w and g to the whole RN +1 by odd reflection with respect to y. Then wodd satisfies the equation − ∆wodd + m2 wodd = godd

in D′ (RN +1 ).

(5.18)

By Lemma 7 we know that godd ∈ Lqloc (RN +1 ) for all q < ∞. Using Theorem 3 in [3] we have wodd ∈ 2,q Wloc (RN +1 ) for all q < ∞. In particular wodd ∈ C 1,α (RN +1 ) for some α ∈ (0, 1). Therefore, w ∈

1,q 2,q +1 +1 +1 +1 ) and so v = wy ∈ Wloc (RN ) ∩ C 0,α (RN ). Wloc (RN ) ∩ C 1,α (RN + + + +

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Using (f1) we deduce that g ∈ C 0,α (RN ). By elliptic boundary regularity for the Dirichlet problem (5.15) +1 +1 we obtain that w ∈ C 2,α (RN ), and hence v = wy ∈ C 1,α (RN ). Finally v satisfies + +  ∇v(x, y)∇η(x, y) + m2 v(x, y)η(x, y)dxdy = 0 (5.19) +1 RN +

+1 +1 for all η ∈ Cc∞ (RN ), so we can conclude that v ∈ C ∞ (RN ). + +



Proof of Theorem 1. This is an immediate consequence of Theorems 7 and 8.



6. Proof of Theorem 2 Consider the following family of functionals depending on m ∈ (0, ω2 )    m 1 2 2 2 2 |∇v| + m v dxdy − |v| dx − F (x, v)dx. Jm (v) = 2 2 ∂ 0 ST ST ∂ 0 ST In the previous section we proved that the functional Jm satisfied the hypotheses of Linking Theorem, in ′ this way we obtained, for all fixed m > 0, the existence of a function vm ∈ Xm T such that Jm (vm ) = 0. In general the estimates obtained on critical levels were dependent on m. Now we want to show that it is possible to take the limit as m → 0 in the problem (1.1). For this we need estimates on critical levels independently of m. Therefore we will prove the following properties: 1. There exist ρ > 0 and λ > 0 independent of m such that inf{Jm (v) : v ∈ Zm = ρ} ≥ λ > 0; T and ∥v∥Xm T

(6.1)

′ m 2. There exists z ∈ Xm T with ∥z∥XT = 1, and there exist R > ρ, δ > 0 independent of m and Rm > 0, such that

max Jm = 0 m ∂AT

and

sup Jm (v) ≤ δ,

(6.2)

Am T

where m ′ Am ≤ Rm , 0 ≤ s ≤ R}. T := {v = y + sz : y ∈ YT , ∥y∥Xm T

We remind that m Xm T = YT −my m where Ym ⟩ and Zm T = ⟨e T = {v ∈ XT : Then, letting



(0,T )N



Zm T

v(x, 0)dx = 0}.

αm := infm max Jm (γ(v)), m γ∈PT v∈AT

where m m PTm = {γ ∈ C 0 (Am T , XT ) : γ = Id on ∂AT },

we deduce by (6.1) and (6.2) that λ ≤ αm ≤ δ for 0 < m < ω2 . We start proving (6.1). Firstly we show that it is possible to obtain the uniform estimates in m for the norm | · |p with 2 ≤ p < 2♯ .

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Let v ∈ Zm T and ε > 0. We denote by ck the Fourier coefficients of the trace of v. Then, taking into account that c0 = 0 and using the trace inequality (ii) of Theorem 3, we have  |v(·, 0)|2L2 (0,T )N = |ck |2 |k|≥1

1   2 2 ω |k| + m2 |ck |2 ω

≤

|k|≥1

1 = |v(·, 0)|2Hm T ω 1 ≤ ∥v∥2Xm . T ω

(6.3)

♯ Now we want to prove that for every v ∈ Zm T and 2 < p < 2

|v(·, 0)|Lp (0,T )N ≤ C∥v∥Xm , T

(6.4)

for some constant C > 0 independent of m. Fix 2 < p < 2♯ and let p′ be its conjugate exponent. Taking into account c0 = 0, (2.10), (2.11) and trace inequality (ii) of Theorem 3 we can see that (here constant C may change from line to line and depend only on T, N, p′ )   1′ ′ p |v(·, 0)|Lp (0,T )N ≤ C |ck |p |k|≥1

   2−p′ ′  p′ 2p − 2−p ′ 2 2 2 ≤ C |v(·, 0)|Hm ( ω |k| + m ) T 

|k|≥1

≤C



p′

− 2−p′

|k|

 2−p′ ′ 2p

|v(·, 0)|Hm T

|k|≥1

≤C



′

|k|

p − 2−p ′

 2−p′ ′ 2p

∥v∥Xm T

(6.5)

|k|≥1 ′ and the last series is finite since N2N +1 < p < 2. Therefore we are able to estimate the functional Jm from below. By Lemma 1 and exploiting (6.3) and (6.4) we can see that for every 0 < m < ω2    m 1 2 2 2 2 |∇v| + m v dxdy − |v| dx − F (x, v)dx Jm (v) = 2 2 ∂ 0 ST ST ∂ 0 ST m  1 ≥ ∥v∥2Xm − + ε |v(·, 0)|2L2 (0,T )N − Cε |v(·, 0)|p+1 Lp+1 (0,T )N T 2 2 1   1 m − + ε ∥v∥2Xm − Cε′ ∥v∥p+1 ≥ Xm T T 2 ω 2 1  ε p+1 2 ′ ≥ − ∥v∥Xm − Cε ∥v∥Xm . T T 4 ω

Choosing 0 < ε < ω4 , we have that b := 41 − ωε > 0. 1   p−1 m Let ρ := 2Cb ′ . Then, for every v ∈ Zm T such that ∥v∥XT = ρ b

Jm (v) ≥ bρ2 − Cb′ ρp+1 =

2 b  b  p−1 =: λ. 2 2Cb′

In this way we can deduce that αm ≥ λ for every 0 < m <

ω 2.

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Now we prove (6.2). Let w(x, y) =

N

i=1

1 and z = sin(ωxi ) y+1

w ∥w∥Xm

We can note that ∥w∥2Xm T ∥w∥2Xm T

=

 T N  2

0

[ 31 ( T2 )N , 13 ( T2 )N −my

and so ∈ Then for every v = ce

∞

2

′

∞

 0

 1 dy (y + 1)2

′′

≡ [C , C ]. Rz, using 0 < m < ω2   µ2 −N µ−2 2 2 ≥T |c + sz(x, 0)| dx

+ω N + sz ∈ Ym T

|v(·, 0)|µLµ (0,T )N

  1 2 2 dy + ω N + m (y + 1)4

2 + ω4 ] 

.

T

(0,T )N

=T

−N

 µ−2 2

(0,T )N

≥ T −N

µ−2 2

≥ T −N

µ−2 2

2



c dx +

(0,T )N

s2 |z(x, 0)|2 dx

 µ2

µ 1  T N 2  2 s T N c2 + ′′ C 2  2 1  T N  µ2 µ (∥y∥2Xm + s2 ) 2 min , ′′ T ω C 2



(6.6)

where in the last inequality we use ∥y∥2Xm = mT N c2 .  T Rz, using (2.18) and (6.6) Hence, if v = ce−my + sz ∈ Ym T   2 s 2 2 Jm (v) = F (x, v)dx ∥z∥Xm − m|z(·, 0)|L2 (0,T )N − T 2 ∂ 0 ST

s2 − (a3 |v(·, 0)|µLµ (0,T )N − a4 T N ) 2 s2 = + a4 T N − a3 |c + sz(·, 0)|µLµ (0,T )N 2  2 1  T N  µ2 µ−2 µ ≤ s2 + a4 T N − a3 T −N 2 min , ′′ (s2 + ∥y∥2Xm ) 2 . T ω C 2 ≤

(6.7)

As a consequence Jm (y + sz) ≤ s2 + a4 T N − a3 C(T, µ)sµ , and so we can find R > ρ such that Jm (y + sz) ≤ 0 for every s ≥ R and y ∈ Ym T . Let 0 ≤ s ≤ R. By (6.7), 2 N ′ ′ . Thus J =: δ for > 0 such that Jm (y + sz) ≤ 0 for every ∥y∥Xm ≥ R there exists Rm m (v) ≤ R + a4 T m T m v ∈ AT . Now we can verify that it is possible to take the limit as m → 0 in (1.1). Fix β ∈ ( µ1 , 12 ) and we proceed ′ as in the first part of Lemma 6. Using the facts that Jm (vm ) = αm ≤ δ for all m ∈ (0, ω2 ) and Jm (vm ) = 0 we have ′ δ ≥ Jm (vm ) − β⟨Jm (vm ), vm ⟩ = 1  = − β [∥vm ∥2Xm − m|vm (·, 0)|2L2 (0,T )N ] T 2 + [βf (x, vm )vm − F (x, vm )]dx ∂ 0 ST  ≥ [βf (x, vm )vm − F (x, vm )]dx

(6.8)

∂ 0 ST

 ≥ (µβ − 1)

∂ 0 ST

F (x, vm )dx − κ

(6.9)

≥ (µβ − 1)[a3 |vm (·, 0)|µLµ (0,T )N − a4 T N ] − κ ≥ (µβ − 1)[a3 |vm (·, 0)|µL2 (0,T )N T −N

µ−2 2

− a4 T N ] − κ.

(6.10)

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By (6.10) we deduce the boundedness in L2 of the trace of vm , that is |vm (·, 0)|L2 (0,T )N ≤ K(δ).

(6.11)

Taking into account Jm (vm ) ≤ δ, (6.11) and (6.9) we obtain ∥∇vm ∥2L2 (ST ) ≤ ∥vm ∥2Xm T

= 2Jm (vm ) + m|vm (·, 0)|2L2 (0,T )N + 2 ≤ 2δ +

 ∂ 0 ST

F (x, vm )dx

ω K(δ) + C(δ) =: K ′ (δ). 2

(6.12)

Moreover, if cm k are the Fourier coefficients of vm (·, 0), by the trace inequality (ii) of Theorem 3 we can see that  2 K ′ (δ) ≥ ∥vm ∥2Xm ≥ |vm (·, 0)|2Hm ≥ ω |k| |cm (6.13) k | , T

T

k∈ZN

and so, using (6.11) we obtain that vm (·, 0) is bounded in HT . Finally we prove that it is possible to estimate vm in L2loc (ST ) by a constant independent of m. Fix α > 0. Since vm ∈ C 1 (ST ) (see Theorem 8), we have that for any x ∈ [0, T ]N and y ∈ [0, α]  y vm (x, y) = vm (x, 0) + ∂y vm (x, s)ds. 0

2

2

2

By using (a + b) ≤ 2a + 2b for all a, b ≥ 0 we obtain 2

2

|vm (x, y)| ≤ 2|vm (x, 0)| + 2



y

0

and applying the H¨ older inequality we deduce   |vm (x, y)|2 ≤ 2 |vm (x, 0)|2 +

y

0

2 |∂y vm (x, s)|ds

  |∂y vm (x, s)|2 ds y .

(6.14)

Integrating (6.14) over (0, T )N × (0, α) and exploiting (6.11) and (6.12) we have ∥vm ∥2L2 ((0,T )N ×(0,α)) ≤ 2α|vm (·, 0)|2L2 (0,T )N + α2 ∥∂y vm ∥2L2 (ST ) ≤ 2αK(δ)2 + α2 K ′ (δ).

Therefore we can extract a subsequence, that for simplicity we denote again with (vm ), and there exists v ∈ L2loc (ST ) such that ∇v ∈ L2 (ST ), vm ⇀ v in L2loc (ST ), ∇vm ⇀ ∇v in L2 (ST ) and vm (·, 0) → v(·, 0) in L2 (0, T )N as m → 0. Now we know that vm satisfies   ∇vm ∇η + m2 vm η dxdy = [mvm + f (x, vm )]η dx (6.15) ∂ 0 ST

ST

∞ for every η ∈ Xm T . Let ϕ ∈ XT and ξ ∈ C ([0, ∞)) such that   if 0 ≤ y ≤ 1  ξ=1 0 ≤ ξ ≤ 1 if 1 ≤ y ≤ 2   ξ=0 if y ≥ 2.

(6.16)

y We set ξR (y) = ξ( R ) for R > 1. Then choosing η = ϕξR ∈ Xm T in (6.15) and taking the limit as m → 0 we have   ∇v∇(ϕξR ) dxdy = f (x, v)ϕ dx. (6.17) ST

∂ 0 ST

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Hence taking the limit as R → ∞ we deduce that v verifies   ∇v∇ϕ dxdy − f (x, v)ϕ dx = 0 ∀ϕ ∈ XT . ∂ 0 ST

ST

Now we want to prove that v ̸≡ 0. Let ξ(y) ∈ C ∞ ([0, ∞)) defined as in (6.16); then ξv ∈ Xm T . Hence ′ 0 = ⟨Jm (vm ), ξv⟩   = ∇vm ∇(ξv) + m2 vm ξv dxdy − m



∂ 0 ST

ST

vm v dx −

∂ 0 ST

f (x, vm )v dx

and as m → 0 we find 

 ∇v∇(ξv) dxdy −

0= ST

∂ 0 ST

f (x, v)v dx.

(6.18)

′ Now, using the facts Jm (vm ) ≥ λ and ⟨Jm (vm ), vm ⟩ = 0, we can see that  2 2λ ≤ 2Jm (vm ) + m|vm (·, 0)|L2 (0,T )N + 2 F (x, vm ) dx ∂ 0 ST  = ∥vm ∥2Xm = m|vm (·, 0)|2L2 (0,T )N + f (x, vm )vm dx ∂ 0 ST

T

and taking the limit as m → 0 we obtain  2λ ≤

∂ 0 ST

f (x, v)v dx.

Taking into account (6.18) and (6.19) we deduce that   0 < 2λ ≤ f (x, v)v dx = ∂ 0 ST

∇v∇(ξv) dxdy

(6.19)

(6.20)

ST

and so v ̸≡ 0. References [1] V. Benci, P.H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (3) (1979) 241–273. URL: http://dx.doi.org/10.1007/BF01389883. ´ B´ [2] A. enyi, T. Oh, The Sobolev inequality on the torus revisited, Publ. Math. Debrecen 83 (3) (2013) 359–374. URL: http://dx.doi.org/10.5486/PMD.2013.5529. [3] H. Brezis, Semilinear equations in RN without condition at infinity, Appl. Math. Optim. 12 (3) (1984) 271–282. URL: http://dx.doi.org/10.1007/BF01449045. [4] X. Cabr´ e, Y. Sire, Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 31 (1) (2014) 23–53. URL: http://dx.doi.org/10.1016/j.anihpc.2013.02.001. [5] X. Cabr´ e, Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc. 367 (2) (2015) 911–941. URL: http://dx.doi.org/10.1090/S0002-9947-2014-05906-0. [6] X. Cabr´ e, J. Sol` a-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (12) (2005) 1678–1732. URL: http://dx.doi.org/10.1002/cpa.20093. [7] X. Cabr´ e, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (5) (2010) 2052–2093. URL: http://dx.doi.org/10.1016/j.aim.2010.01.025. [8] L.A. Caffarelli, J.-M. Roquejoffre, Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (5) (2010) 1151–1179. URL: http://dx.doi.org/10.4171/JEMS/226. [9] L.A. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2) (2008) 425–461. URL: http://dx.doi.org/10.1007/s00222-007-0086-6. [10] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (7–9) (2007) 1245–1260. URL: http://dx.doi.org/10.1080/03605300600987306. [11] R. Carmona, W.C. Masters, B. Simon, Relativistic Schr¨ odinger operators: asymptotic behavior of the eigenfunctions, J. Funct. Anal. 91 (1) (1990) 117–142. URL: http://dx.doi.org/10.1016/0022-1236(90)90049-Q.

284

V. Ambrosio / Nonlinear Analysis 120 (2015) 262–284

[12] V. Coti Zelati, M. Nolasco, Existence of ground states for nonlinear, pseudo-relativistic Schr¨ odinger equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22 (1) (2011) 51–72. URL: http://dx.doi.org/10.4171/RLM/587. [13] J. Fr¨ ohlich, B.L.G. Jonsson, E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys. 274 (1) (2007) 1–30. URL: http://dx.doi.org/10.1007/s00220-007-0272-9. [14] J. Fr¨ ohlich, E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math. 60 (11) (2007) 1691–1705. URL: http://dx.doi.org/10.1002/cpa.20186. [15] J. Fr¨ ohlich, E. Lenzmann, Dynamical collapse of white dwarfs in Hartree- and Hartree–Fock theory, Comm. Math. Phys. 274 (3) (2007) 737–750. URL: http://dx.doi.org/10.1007/s00220-007-0290-7. [16] W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to a semilinear Schr¨ odinger equation, Adv. Differential Equations 3 (3) (1998) 441–472. [17] E.H. Lieb, M. Loss, Analysis, second ed., in: Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. [18] E.H. Lieb, H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys. 112 (1) (1987) 147–174. URL: http://projecteuclid.org/euclid.cmp/1104159813. [19] E.H. Lieb, H.-T. Yau, The stability and instability of relativistic matter, Comm. Math. Phys. 118 (2) (1988) 177–213. URL: http://projecteuclid.org/euclid.cmp/1104161986. [20] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, in: Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989, URL: http://dx.doi.org/10.1007/978-1-4757-2061-7. [21] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS Regional Conference Series in Mathematics, vol. 65, The American Mathematical Society, Providence, RI, 1986, Published for the Conference Board of the Mathematical Sciences, Washington, DC. [22] M. Ryznar, Estimates of Green function for relativistic α-stable process, Potential Anal. 17 (1) (2002) 1–23. URL: http://dx.doi.org/10.1023/A:1015231913916. [23] R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (5) (2013) 2105–2137. [24] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (1) (2007) 67–112. URL: http://dx.doi.org/10.1002/cpa.20153. [25] Y. Sire, E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal. 256 (6) (2009) 1842–1864. URL: http://dx.doi.org/10.1016/j.jfa.2009.01.020. [26] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, fourth ed., in: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics ([Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]), vol. 34, Springer-Verlag, Berlin, 2008. [27] M. Willem, Minimax Theorems, in: Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkh¨ auser Boston, Inc., Boston, MA, 1996, URL: http://dx.doi.org/10.1007/978-1-4612-4146-1. [28] A. Zygmund, Trigonometric Series. Vol. I, II, third ed., in: Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002, With a foreword by Robert A. Fefferman.