DOI 10.1007/s00526-006-0078-1
Calculus of Variations
Brake orbits type solutions to some class of semilinear elliptic equations
Francesca Alessio · Piero Montecchiari
Received: 8 April 2006 / Accepted: 10 October 2006 / Published online: 10 January 2007
© Springer-Verlag 2007
Abstract We consider a class of semilinear elliptic equations of the form
−u(x, y) + a(x)W(u(x, y)) = 0, (x, y) ∈R2 (0.1) where a : R → Ris a periodic, positive function and W : R → Ris modeled on the classical two well Ginzburg-Landau potential W(s) = (s2− 1)2. We show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem
−¨q(x) + a(x)W(q(x)) = 0, x ∈R, q(±∞) = ±1,
has a discrete structure, then (0.1) has infinitely many solutions periodic in the variable y and verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y∈R.
Mathematics Subject Classification (2000) 35J60· 35B05 · 35B40 · 35J20 · 34C37
1 Introduction
We consider semilinear elliptic equations of the form
−u(x, y) + a(x)W(u(x, y)) = 0, (x, y) ∈R2 (1.1) where we assume
Supported by MURST Project ‘Metodi Variazionali ed Equazioni Differenziali Non Lineari’.
F. Alessio (B)· P. Montecchiari
Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy
e-mail: alessio@dipmat.univpm.it P. Montecchiari
e-mail: montecch@mta01.univpm.it
(H1) a :R→Ris Hölder continuous, T-periodic and satisfies 0< a = minx∈Ra(x) <
maxx∈Ra(x) ≡ a;
(H2) W ∈ C2(R) satisfies W(s) ≥ 0 for any s ∈ R, W(s) > 0 for any s ∈ (−1, 1), W(±1) = 0 and W(±1) > 0.
Potentials satisfying the assumption(H2) are widely used in physical models. We refer for example to the Ginzburg-Landau potential, W(s) = (s2− 1)2, and to the Sine-Gordon potential, W(s) = 1 + cos(πs), introduced to study various problems in phase transitions and condensed state Physics. The global minima of W represent in these models pure phases, energetically favorite, of the material. The introduction of an oscillatory factor a can be used to describe inhomogeneity of the material.
Pursuing the study already faced in [4,6,7] we consider in this paper the problem of multiplicity of solutions to the boundary value problem
⎧⎪
⎪⎨
⎪⎪
⎩
−u(x, y) + a(x)W(u(x, y)) = 0, (x, y) ∈R2,
|u(x, y)| ≤ 1, (x, y) ∈R2,
x→±∞lim u(x, y) = ±1, uniformly w.r.t. y∈R.
(1.2)
As it is nowaday well known, the multiplicity problem can have solution only if the oscillatory function a is not constant. Indeed, in [18] Ghoussoub and Gui proved in the planar case a long standing conjecture by Ennio De Giorgi [16] (see also [2,9,11,12,17]
where the results in [18] are extended to higher dimensions and to more general set- tings; see finally [24] where the conjecture has been proved inRnfor n ≤ 8). As a particular consequence of Their work we know that in the autonomous case, i.e. when a(x) = a0 > 0 for any x ∈ R, if u is a solution of (1.2), then u(x, y) = q(x) for all (x, y) ∈R2, where q is a solution of the one dimensional problem associated to (1.2)
⎧⎪
⎨
⎪⎩
−¨q(x) + a(x)W(q(x)) = 0, x ∈R,
|q(x)| ≤ 1, a∈R,
limx→±∞q(x) = ±1.
(1.3)
Furthermore, when a is constant, we know that the set of solutions of (1.3) is a contin- uum homeomorphic toR, being constituted by the translations of a single heteroclinic solution q0. These results tell us that for autonomous equations the set of the solutions to (1.2) reduces, modulo space translation, to this single one dimensional solution q0
and the problem (1.2) is in fact one dimensional.
As shown in [4], when a is not constant the one dimensional symmetry of the problem disappears even if the potential depends only on the single variable x, as in (1.2) (for the case of a depending on both the space variable and for related prob- lems see e.g. [5,10,14,15,20–23] and the references therein). Related to [4] are the papers by Alama et al. [1], and by Schatzman [25], where the Authors study systems of autonomous Allen Cahn equations which admit two dimensional solutions. In all these works the construction of two dimensional solutions is done assuming that the set of one dimensional solutions, differently from the autonomous equation case, has a discrete structure.
In the present paper we study the problem (1.2) under the same discreteness assumption on the set of one dimensional solutions made in [4]. To be more precise
let us define the action functional F(q) =
R 1
2|˙q(x)|2+ a(x)W(q(x)) dx
on the class = z0+ H1(R), where z0∈ C∞(R) is any fixed increasing function such that z0(x) → ±1 as x → ±∞ and |z0(x)| = 1 for any |x| ≥ 1. Setting c = infF we consider the set of minimal one dimensional solutions of (1.2):K= {q ∈ / F(q) = c}.
We assume the following
(∗) there exists ∅ =K0⊂Ksuch that, settingKj= {q(· − jT) | q ∈K0} for j ∈Z, there results
(i) K0is compact with respect to the H1(R) topology,
(ii) K= ∪j∈ZKjand there exists d0> 0 such that if j = jthend(Kj,Kj) ≥ d0
whered(A, B) = inf{ q1(x) − q2(x) L2(R)/ q1 ∈ A, q2∈ B}, A, B ⊂ .
We remark that the assumption(∗) excludes the autonomous case, since, when a is constant, the problem is invariant under the continuous group of translations and (ii) cannot hold. On the other hand, Melnikov–Poincaré methods (see [8] and the references therein) allows to check(∗) when a is a small L∞periodic perturbation of a positive constant. We refer moreover to [4] where, following [3],(∗) is verified when a is a slowly oscillating function (in particular, is satisfied for the equation
−ε2u + a(x)W(u) = 0 when ε > 0 is small enough), and more generally when the function a is a slowly oscillating perturbations of an arbitrary periodic functions. For a more general study of “gap” conditions of this kind we finally refer to the recent paper by Matano and Rabinowitz [19].
The existence of solutions depending on both the planar variables is achieved in [4] prescribing different asymptotic behaviours of the solutions as y→ ±∞. Precisely solutions u to problem (1.2) which belong to the space
H=
u∈ H1loc(R2) / u L∞(R2)≤ 1 and u − z0∈ ∩(ζ1,ζ2)⊂RH1(R× (ζ1,ζ2)) , and which are asymptotic as y→ ±∞ to different minimal setsKj−,Kj+are found.
To have a different view of that result we can adopt a terminology of the Dynam- ical Systems Theory. Indeed, roughly speaking, any solution u∈Hof (1.2) defines a trajectory y∈R→ u(·, y) ∈ , solution of the infinite dimensional dynamical system
d2
dy2u(·, y) = F(u(·, y)).
If we think at the variable y as a time variable and at the functional−F as a potential on we recognize a very simple Lagrangian structure. The one dimensional solutions of (1.2) are equilibria of the system and the two dimensional solutions found in [4] are heteroclinic solutions to the system connecting as y→ ±∞ different maxima (Kj±) of the potential−F.
Note also that the system is autonomous, i.e. its Lagrangian does not explicitly depend on y. This implies, as shown in [6], that if u∈His a solution to (1.2) then the Energy function
y→ Eu(y) = −1
2 ∂yu(·, y) 2L2(R)+ F(u(·, y))
is constant. In particular the heteroclinic solutions found in [4] have Energy equal to c.
In the present paper we extend the result in [4] proving the existence of infinitely many periodic solutions of (1.2) of the brake orbits type.
We recall (see [26]) that for an autonomous Lagrangian system of the type−¨q(t) = V(q(t)), (q ∈Rn) a solution q is said to be a brake orbit solution on{V < E}, if it has Energy E and there exists T> 0 such that, up to translations,
V(q(0)) = V(q(T)) = E and V(q(t)) < E for any t ∈ (0, T).
By conservation of Energy we have˙q(0) = ˙q(T) = 0 and the solution can be continued toRaccording to the formulas q(−t) = q(t) and q(T + t) = q(T − t). That produces a periodic solution of period 2T which oscillates back and forth in the configuration space along a simple curve connecting the two turning points q(0) and q(T).
For our problem, looking for a brake orbit type solution, means to look for a level cp> c a real number Tp> 0 and a solution vp∈Hof (1.2) with Energy Evp(y) = cp, which satisfies
F(vp(·, 0)) = F(vp(·, Tp)) = cp, F(vp(·, y)) > cp for any y∈ (0, Tp) and the symmetry properties
vp(·, −y) = vp(·, y) and vp(·, T + y) = vp(·, T − y) for any y ∈R. To precise our result we observe that the discreteness of the minimal setKreflects on the structure of the sublevels of F and we have that there exists c∗> c such that
q∈ / F(q)≤c∗
= ∪j∈Zj, withKj⊂j for any j∈Z and inf
i=jd(i,j)>0.
We prove the following
Theorem 1.1 Let a and W satisfy(H1), (H2) and assume that condition (∗) holds true.
If cp∈ (c, c∗) is a regular value of F then there exists Tp> 0, jp∈Z\ {0} and a solution vp∈C2(R2) to problem (1.2) such that
(i) Evp(y) = −12 ∂yvp(·, y) 2L2(R)+ F(vp(·, y)) = cpfor any y∈R,
(ii) F(vp(·, 0)) = F(vp(·, Tp)) = cp, vp(·, 0) ∈ 0, vp(·, Tp) ∈ jpand F(vp(·, y)) > cp
for any y∈ (0, Tp),
(iii) vp(·, −y) = vp(·, y) and vp(·, y + Tp) = vp(·, Tp− y) for any y ∈R. In particular vp(x, y + 2Tp) = vp(x, y) for any (x, y) ∈R2.
We remark that, by conservation of Energy, the solution vpsatisfies the Neumann boundary conditions∂yvp(x, 0) = ∂yvp(x, Tp) = 0 for any x ∈ R. Moreover since vp(·, 0) ∈ 0, vp(·, Tp) ∈ jpand jp= 0, vpis a two dimensional solution of (1.2).
Our second consideration regards the multiplicity result contained in Theorem 1.1. Indeed the Theorem guarantees the existence of a brake orbit type solution at level cpwhenever cp ∈ (c, c∗) is a regular value of F. As a consequence of the Sard Smale Theorem and the local compactness properties of F, we prove in Lemma2.9 below that the set of regular values of F is open and dense in[c, c∗]. Then, Theorem 1.1provides in fact the existence of an uncountable set of geometrically distinct two dimensional solutions of (1.2) of the brake orbit type.
To prove Theorem1.1we make use of variational methods and we apply an Energy constrained variational Principle already introduced and used in [6,7]. Given cp ∈ (c, c∗), regular for F, we look for minima of the renormalized functional
ϕp(u) =
R 1
2 ∂yu(·, y) 2L2(R)+ (F(u(·, y)) − cp) dy on the space
Mp=
u∈H/ lim
y→−∞d(u(·, y), 0)=0, lim inf
y→+∞d(u(·, y), 0)>0 and inf
y∈RF(u(·, y))≥cp
. Thanks to the constraint infy∈RF(u(·, y)) ≥ cp, the functionalϕpis well defined onMp. Moreover its minimizing sequences admit limit points up∈H(a priori not inMp) with respect to the weak topology of Hloc1 (R2). Defining sp= sup{y ∈R/d(up(·, y), 0) = 0 and F(up(·, y)) ≤ cp} and tp= inf{y > sp/ F(up(·, y)) ≤ cp}, we have up(·, sp) ∈ 0, up(·, tp) ∈ jpand F(up(·, y)) > 0 for any y ∈ (sp, tp). Then, the minimality properties of upallow us to prove that
(a) upsolves in a classical sense the equation−u + aW(u) = 0 onR× (sp, tp), (b) Eup(y) = −12 ∂yup(·, y) 2L2(R)+ F(up(·, y)) = cpfor any y∈ (sp, tp).
Finally, since cpis regular for F, there results that−∞ < sp < tp< +∞ and setting Tp = tp− sp, we recover the brake orbit solution vp by translating, reflecting and periodically continuing the function up.
The detailed analysis of the above sketched arguments is made in Sect. 3 while the Sect. 2 is devoted to recall a series of preliminary results concerning variational aspects of the one dimensional problem.
2 The one dimensional problem
In this section we recall some results concerning the one dimensional problem asso- ciated to (1.2). We focus in particular our study on some variational properties of the one dimensional solutions to (1.2), i.e. solutions to the heteroclinic problem
−¨q(x) + a(x)W(q(x)) = 0, x ∈R,
x→±∞lim q(x) = ±1. (2.1)
We fix any function z0∈ C∞(R) such that z0(x) → ±1 as x → ±∞ and |z0(x)| = 1 for any|x| ≥ 1. Then, we consider on the space
= z0+ H1(R), the functional
F(q) = 1
2 ˙q 2L2(R)+
R
a(x)W(q(x)) dx.
Remark 2.1 We note that F is weakly lower semicontinuous with respect to the H1loc(R) convergence. Moreover, endowing with the hilbertian structure induced by the map p : H1(R) → , p(z) = z0+ z, it is classical to prove that F ∈ C2() with Frechet differential
F(q)h = ˙q, ˙hL2(R)+
R
a(x)W(q(x))h(x) dx, q ∈ , h ∈ H1(R),
and that critical points of F are classical solutions to (2.1).
We are interested in the minimal properties of F on and we set c= inf
F and K= {q ∈ / F(q) = c} . Moreover, given any ∈Rwe denote{F ≤ } = {q ∈ | F(q) ≤ }.
If I is an interval inR, we set FI(q) =
I
1
2|˙q|2+ a(x)W(q) dx,
noting that FI(q) is well defined on H1loc(R), weakly lower semicontinuous, with values in[0, +∞].
Finally, for a given q ∈ L2(R) we let q ≡ q L2(R) and given A, B ⊂ L2(R) we denote
d(A, B) = inf { q1− q2 | q1∈ A, q2∈ B} .
Remark 2.2 We precise some basic consequences of the assumptions(H1) and (H2), fixing some constants which will remain unchanged in the following.
First, we note that, by(H2), there exists δ ∈ (0,14) and w > w > 0 such that
w≥ W(s) ≥ w for any |s| ∈ [1 − 2δ, 1 + 2δ]. (2.2) In particular, since W(±1) = W(±1) = 0, setting χ(s) = min{|1 − s|, |1 + s|}, we have that
if|s| ∈ [1 − 2δ, 1 + 2δ], thenw
2χ(s)2≤ W(s) ≤ w
2χ(s)2and|W(s)| ≤ wχ(s). (2.3) Note that by (2.3) we derive the existence of two constants b, b> 0 such that
bχ(s)2≤ W(s) and |W(s)| ≤ b χ(s), ∀|s| ≤ 1 + 2δ. (2.4) Moreover, denoting
ωδ= min
|s|≤1−δW(s), δ ∈ (0, 1), (2.5)
we have thatωδ> 0 for any δ ∈ (0, 1) and in particular λ0≡ 1
2min
2aωδδ ; 2a bδ2
> 0. (2.6)
In the sequel we will study some properties of the sublevel set{F ≤ c + λ0}. First of all we note that if q∈ Hloc1 (R) is such that W(q(x)) ≥ µ > 0 for any x ∈ (σ , τ) ⊂R, then
F(σ ,τ)(q) ≥ 1
2(τ − σ )|q(τ) − q(σ )|2+ aµ(τ − σ ) ≥
2aµ |q(τ) − q(σ )|. (2.7) As first consequence we obtain the following estimate
Lemma 2.1 If q∈ {F ≤ c + λ0} then q L∞(R)≤ 1 + 2δ.
Proof Arguing by contradiction, assume that q ∈ and there exists x0 such that
|q(x0)| > 1 + 2δ. If q(x0) > 1 + 2δ (analogous is the case q(x0) < −1 − 2δ) then, by continuity, there exists x1< σ < τ ∈Rsuch that q(x1) = 1, q(σ ) = 1+δ, q(τ) = 1+2δ and 1+δ < q(x) < 1+2δ for any x ∈ (σ , τ). By (2.4), W(q(x)) ≥ bδ2for any x∈ (σ , τ).
Then, by (2.7) and the definition ofλ0given in (2.6), we obtain F(σ ,τ)(q) ≥ 2λ0. More- over, since q∈ and q(x1) = 1, we have also F(−∞,x1)(q) ≥ c. Then, we reach the contradiction c+ λ0≥ F(q) ≥ F(−∞,x1)(q) + F(σ ,τ)(q) ≥ c + 2λ0.
As a consequence of Lemma2.1we plainly obtain the following
Lemma 2.2 Let(qn) ⊂ {F ≤ c + λ0}. Then, there exists q ∈ H1loc(R) with q L∞(R)≤ 1+ 2δ such that, along a subsequence, qn → q weakly in Hloc1 (R), ˙qn→ ˙q weakly in L2(R) and moreover F(q) ≤ lim infn→∞F(qn).
Proof Since F(qn) ≤ c+λ0for any n∈N, by Lemma2.1we recover that qn L∞(R)≤ 1+ 2δ. Since moreover ˙qn ≤ 2(c + λ0) for any n ∈Nwe obtain that there exists q∈ H1loc(R) with q L∞(R)≤ 1 + 2δ such that, along a subsequence, qn→ q weakly in Hloc1 (R), ˙qn→ ˙q weakly in L2(R). Then, the Lemma follows by Remark2.1.
We note now that by a simple comparison argument, one easily recognizes that if a function q∈ Hloc1 (R) verifies q(σ ) = −1 + δ and q(τ) = 1 − δ for certain σ < τ ∈R andδ ∈ (0, 1), then F(σ ,τ)(q) ≥ c − oδwith oδ→ 0 as δ → 0. In particular, in relation with the constantλ0fixed in (2.6) we have
Lemma 2.3 There exists δ0 ∈ (0, δ) such that if q ∈ verifies q(σ ) = −1 + δ and q(τ) = 1 − δ for some σ < τ ∈Randδ ∈ (0, δ0], then F(σ ,τ)(q) ≥ c −λ80.
For a given q∈ we define
σq= sup {x ∈R/ q(x) ≤ −1 + δ0} and τq= inf
x> σq/ q(x) ≥ 1 − δ0
. Since q(x) → ±1 as x → ±∞ and q is continuous, we clearly have σq< τq∈Rand
q(σq) = −1 + δ0, q(τq) = 1 − δ0 and |q(x)| < 1 − δ0 for all x∈ (σq,τq).
(2.8) Moreover there results
Lemma 2.4 For every q∈ {F ≤ c + λ0} we have (i) τq− σq≤ L0≡ c+λaωδ00.
(ii) if x< σq, then−1−2δ ≤ q(x) ≤ −1+2δ and if x > τq, then 1−2δ ≤ q(x) ≤ 1+2δ.
Proof By (2.7), (2.8) and (2.5) we have F(σq,τq)(q) ≥ aωδ0(τq − σq). Then, since F(σq,τq)(q) ≤ F(q) ≤ c + λ0, (i) plainly follows.
To show that (ii) holds true, assume by contradiction (using Lemma2.1) that there existsσ < σqsuch that q(σ ) > −1 + 2δ or τ > τq such that q(τ) < 1 − 2δ. Since
|q(σq)| = |q(τq)| = 1 − δ0 > 1 − δ, in both the cases we have that there exists an interval (x−, x+) ⊂ R\ (σq,τq) such that |q(x)| ≤ 1 − δ for any x ∈ (x−, x+) and
|q(x+) − q(x−)| = δ. Then, by (2.5), we have W(q(x)) ≥ ω¯δfor any x∈ (x−, x+) and hence, by (2.7) and the definition ofλ0 given in (2.6), we get F(x−,x+)(q) ≥ 2λ0. By Lemma2.3we conclude c+ λ0 ≥ F(q) ≥ F(σq,τq)(q) + F(x−,x+)(q) ≥ c − λ80 + 2λ0, a
contradiction which proves (ii).
The concentration property of the functions q∈ {F ≤ c + λ0} described in Lemma 2.4allows us to obtain the following compactness result.
Lemma 2.5 Let(qn) ⊂ {F ≤ c + λ0} be such that the sequence (σqn) is bounded inR. Then, there exists a subsequence(qnk) ⊂ (qn) and q ∈ such that qnk− q → 0 weakly in H1(R). Moreover, if F(qnk) → F(q) then qnk− q → 0 strongly in H1(R).
Proof By Lemma2.2there exists a subsequence(qnk) ⊂ (qn), q ∈ Hloc1 (R) such that
˙q ∈ L2(R), q L∞(R)≤ 1 + 2δ, qnk→ q weakly in Hloc1 (R), ˙qnk → ˙q weakly in L2(R).
To prove the first part of the Lemma we have to show that q∈ and that qnk− q → 0 weakly in L2(R).
To this aim note that, since the sequence(σqn) is bounded inRand(qn) ⊂ {F ≤ c+λ0}, by Lemma2.4, there exists T0> 0 such that for any n ∈N
if x< −T0then−1−2δ ≤ qn(x) ≤ −1+2δ and if x > T0then 1−2δ ≤ qn(x) ≤ 1+2δ.
(2.9) By the L∞locconvergence, (2.9) holds true even for the function q
if x< −T0then − 1 − 2δ ≤ q(x) ≤ −1 + 2δ and if x > T0then 1− 2δ ≤ q(x) ≤ 1 + 2δ.
(2.10) Then, by (2.3) and (2.10), we have
x<−T0
|q + 1|2=
x<−T0
χ(q)2 ≤ 2 w
x<−T0
W(q) ≤ 2
a wF(q) ≤ 2(c + λ0) a w
and analogously
x>T0|q − 1|2≤ 2(c+λa w0). Since we already know that ˙q ∈ L2(R), this implies that q− z0∈ H1(R), i.e., q ∈ .
By (2.3) and (2.9) we recover that also
|x|>T0χ(qn)2≤ 4(c+λa w0)for any n∈Nand so, by Lemma2.1, the sequence qn− q is bounded. This implies, as we claimed, that qnk− q → 0 weakly in L2(R).
To prove the second part of the Lemma, note that since qnk → q in L∞loc(R) and
˙qnk→ ˙q weakly in L2(R), for any T ≥ T0we have F(qnk) − F(q) = 1
2 ˙qnk− ˙q 2+
|x|>T
a(W(qnk) − W(q)) dx + o(1) as k → ∞
and since by assumption we have F(qnk) → F(q), we derive that 1
2 ˙qnk− ˙q 2+
|x|>T
aW(qnk) dx =
|x|>T
aW(q) dx + o(1) as k → ∞. (2.11)
By (2.3) and (2.9), we know that for any|x| ≥ T0
a(x)W(qnk(x)) ≥ a wχ(qnk(x))2 and hence, by (2.11) we obtain
1
2 ˙qnk− ˙q 2+ a w
|x|>T
χ(qnk)2dx≤
|x|≥T
aW(q) dx + o(1) as k → ∞. (2.12)
Then, since by (2.9) and (2.9), for|x| ≥ T0we have|qnk(x) − q(x)|2≤ 2(χ(qnk(x))2+ χ(q(x))2) for any k ∈N, and since
RaW(q) dx < +∞,
Rχ(q)2dt< +∞, by (2.12)
we obtain that for anyη > 0 there exists Tη> T0such that as k→ +∞
1
2 ˙qnk− ˙q 2+ a w
|x|>Tη
|qnk− q|2dx≤ 1
2 ˙qnk− ˙q 2+ 2a w
|x|>Tη
χ(qnk)2+ χ(q)2dx
≤ 2
|x|>Tη
aW(q) dx + 2a w
|x|>Tη
χ(q)2dx+ o(1) ≤ η + o(1).
Since qnk− q → 0 in L∞loc(R), this implies qnk− q H1(R)→ 0 as k → ∞.
By Lemma2.5, we derive compactness properties of the minimizing sequences of F in. Given x ∈R, we denote with[x] the entire part of x.
Lemma 2.6 Let(qn) ⊂ be such that F(qn) → c. Then, there exists q ∈Ksuch that, along a subsequence, qn(· + [σqn]) − q H1(R)→ 0 as n → ∞.
Proof We note that setting vn= qn(· + [σqn]), by 1-periodicity of the function a, we have F(vn) = F(qn) → c as n → +∞ and moreover σvn ∈ [0, 1). By Lemma2.5we obtain that there exist q∈ and a subsequence of (vn), still denoted (vn), such that vn− q → 0 weakly in H1(R) and, by Remark2.1, F(q) ≤ c. Then, since q ∈ , we derive F(q) = c, i.e., q ∈K. Finally, since F(q) = c = limn→∞F(vn), by Lemma2.5, we obtain vn− q H1(R)→ 0 as n → ∞, and the Lemma follows.
Remark 2.3 By Lemma2.6in particular we obtain thatK= ∅ and by standard argu- ment, we have that every q∈Kis a classical solution to problem (2.1). Moreover,
(i) By Lemma2.6, using a contradiction argument, we obtain that for any r > 0 there existsνr> 0 such that
if q∈ {F ≤ c + λ0} and inf
¯q∈K q − ¯q H1(R)≥ r then F(q) ≥ c + νr. (2.13) (ii) If q1, q2 ∈Kandσq1 = σq2then q1 = q2. Indeed, by (2.2), for any givenσ ∈R,
the functional F(−∞,σ )is strictly convex on the convex set Gσ =
q∈ −1 + H1((−∞, σ )) / q(σ ) = −1 + δ0and − 1 − 2δ ≤ q(x)
≤ −1 + 2δ for any x ≤ σ ,
and we derive that there exists a unique Qσ ∈ Gσ such that F(−∞,σ )(Qσ) = min{F(−∞,σ )(q) / q ∈Gσ}. Then since q1and q2are minima of F on and since σq1 = σq2 ≡ σ , by Lemma2.4both the functions q1|(−∞,σ ) and q2|(−∞,σ ) are minima of F(−∞,σ ) onGσ and so coincide with the function Qσ. Therefore, by uniqueness of the solution of the Cauchy problem we conclude that q1= q2. (iii) If q∈Kthen q L∞(R)≤ 1.
Indeed, letσ = inf{x ∈ R/ q(x) ≥ −1} and τ = sup{x ∈ R/ q(x) ≤ 1}, then
|q(x)| ≤ 1 for all x ∈ (σ , τ). Assume by contradiction that (σ , τ) = R and consider the function ˜q(x) = max{min{q(x); 1}; −1}. Then, since by Lemma2.1, q L∞(R)≤ 1 + 2¯δ, by (2.4) we obtain F(q) > F(˜q) which is a contradiction since F(q) = c and ˜q ∈ .
By Remark2.3(ii), we see that the map q → σq is injective onK. In the next Lemma we will prove that the map is continuous onKwith continuous inverse and so that the setKis homeomorphic to the real set
S(K) =
σ ∈R/ there exists q ∈Ksuch thatσq= σ .
