Positive semiclassical states for a fractional Schr\"odinger-Poisson system

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by  Edwin G. Murcia

POSITIVE SEMICLASSICAL STATES FOR A
FRACTIONAL SCHRO¨DINGER-POISSON SYSTEM
EDWIN GONZALOMURCIA ANDGAETANO SICILIANO
6
1
0 Abstract. We consider a fractional Schr¨odinger-Poisson system in the whole space RN in
2
presence of a positive potential and depending on a small positive parameter ε. We show

n that, for suitably small ε (i.e. in the “semiclassical limit”) the number of positive solutions
a is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the
J
potential.

4

] 1. Introduction
P
A
In the last decades a great attention has been given to the following Schro¨dinger-Poisson
.
h type system
at −∆u+V(x)u+φu= |u|p−2u
m −∆φ= u2,
(cid:26)
[
which arises in non relativistic Quantum Mechanics. Such a system is obtained by looking for

1 standing waves solutions in the purely electrostatic case to the Schro¨dinger-Maxwell system.
v For a deduction of this system, see e.g. [12]. Here the unknownsare u, the modulusof thewave
5
function, and φ which represents the electrostatic potential. V is a given external potential
8
4 and p ≥ 2 a suitable given number.
0 The system has been studied by many authors, both in bounded and unbounded
0
domains, with different assumptions on the data involved: boundary conditions, potentials,
.
1 nonlinearities; many different type of solutions have been encountered (minimal energy, sign
0
changing, radial, nonradial...), the behaviour of the solutions (e.g. concentration phenomena)
6
1 has been studied as well as multiplicity results have been obtained. It is really difficult to give
: a complete list of references: the reader may see [13] and the references therein.
v
i However it seems that results relating the number of positive solutions with topological
X
invariants of the “objects” appearing in the problem are few in the literature. We cite the
ar paper [34] where the system is studied in a (smooth and) bounded domain Ω ⊂ R3 with
u = φ = 0 on ∂Ω and V constant. It is shown, by using variational methods that, whenever p
issufficientlynearthecriticalSobolevexponent6, thenumberofpositivesolutionsisestimated
below by the Ljusternick-Schnirelamnn category of the domain Ω.
On the other hand it is known that a particular interest has the semiclassical limit of the
Schro¨dinger-Poisson system (that is when the Plank constant ~ appearing in the system, see
e.g. [12], tends to zero) especially due to the fact that this limit describes the transition from
Quantum to Classical Mechanics. Such a situation is studied e.g. in [33], among many other
papers. We cite also Fang and Zhao [23] which consider the following doubly perturbedsystem
2000 Mathematics Subject Classification. 35J50, 35Q40, 58E05,
Key words and phrases. Fractional Laplace equation, multiplicity of solutions, Ljusternick-Schnirelmann
category.
EdwinG.MurciaissupportedbyCapes,Brazil. GaetanoSicilianoissupportedbyFapespandCNPq,Brazil.
1
2 E.G.MURCIAANDG.SICILIANO
in the whole space R3:
−ε2∆w+V(x)w+ψw = |w|p−2w
−ε∆ψ = w2.
(cid:26)
Here V is a suitable potential, 4 < p < 6, and ε is a positive parameter proportional to ~.
In this case the authors estimate, whenever ε tends to zero, the number of positive solutions
by the Ljusternick-Schnirelamnn category of the set of minima of the potential V, obtaining a
result in the same spirit of [34].
Recently, especially after the formulation of the Fractional Quantum Mechanics, the
derivation of the Fractional Schro¨dinger equation given by N. Laskin in [27–29], and the notion
of fractional harmonic extension of a function studied in the pioneering paper [15], equations
involving fractional operators are receiving a great attention. Indeed pseudodifferential
operators appear in many problems in Physics and Chemistry, see e.g. [30,31]; but also in
obstacle problems [32,35], optimization and finance [20], conformal geometry and minimal
surfaces [14,16,17], etc.
Motivated by the previous discussion, we investigate in this paper the existence of positive
solutions for the following doubly singularly perturbed fractional Schro¨dinger-Poisson system
in RN:
ε2s(−∆)sw+V(x)w+ψw = f(w)
(P )
ε ( εθ(−∆)α/2ψ = γαw2,
πN/22αΓ(α/2)
where γ := is a constant (Γ is the Euler function). By a positive solution of (P )
α Γ(N/2−α/2) ε
we mean a pair (w,ψ) where w is positive. To the best of our knowledge, there are only few
recent papers dealing with a system like (P ): in [37] the author deals with ε = 1 proving
ε
under suitable assumptions on f the existence of infinitely many (but possibly sign changing)
solutions by means of the Fountain Theorem. A similar system is studied in [36] and the
existence of infinitely many (again, possibly sign changing) solutions is obtained by means of
the Symmetric Mountain Pass Theorem.
In this paper we assume that
(H1) s ∈ (0,1), α ∈ (0,N), θ ∈ (0,α), N ∈ (2s,2s+α),
moreover the potential V and the nonlinearity f satisfy the assumptions listed below:
(V1) V : RN → R is a continuous function and
0 < minV := V < V := liminf V ∈ (V ,+∞];
0 ∞ 0
RN |x|→+∞
(f1) f : R → R is a function of class C1 and f(t)= 0 for t ≤ 0;
(f2) lim f(t)/t = 0;
t→0
(f3) there is q ∈ (2,2∗ −1) such that lim f(t)/tq0 = 0, where 2∗ := 2N/(N −2s);
0 s t→∞ s
t
(f4) there is K > 4 such that 0 < KF(t) := K f(τ)dτ ≤ tf(t) for all t > 0;
0
(f5) the function t 7→ f(t)/t3 is strictly increasiRng in (0,+∞).
The assumptions on the nonlinearity f are quite standard in order to work with variational
methods, use the Nehari manifold and the Palais-Smale condition. The assumption (V1) will
be fundamental in order to estimate the number of positive solutions and also to recover some
compactness.
FRACTIONAL SCHRO¨DINGER-POISSON SYSTEM 3
We recall, once for all, that a C1 functional J, defined on a smooth manifold M, is said to
satisfythePalais-Smaleconditionatlevelc ∈ R((PS) forbrevity)ifeverysequence{u } ⊂ M
c n
such that
(1.1) J(u )→ c and J′(u )→ 0
n n
hasaconvergentsubsequence. Asequence{u }satisfying(1.1)isalsonameda(PS) sequence.
n c
To stay our result let us introduce
M := x ∈ RN :V(x) =V
0
n o
the set of minima of V. Our result is the following
Theorem 1.1. Under the above assumptions (H1), (V1), (f1)-(f5), there exists an ε∗ > 0 such
that for every ε∈ (0,ε∗] problem (P ) possesses at least catM positive solutions.
ε
Moreover if catM > 1 and M is bounded, then (for suitably small ε) there exist at least
catM +1 positive solutions.
Hereafter, given atopological pair (X,Y),cat (Y) is theLjusternick-Schnirelmanncategory
X
of Y in X, and, if X = Y this is just denoted with catX.
The proof of Theorem 1.1 is carried out by adapting some ideas of Benci, Cerami and
Passaseo [10,11] and using the Ljusternick-Schnirelmann Theory. We mention that these
ideas and techniques have been extensively used to attack also other type of problems, and
indeed similar results are obtained for other equations and operators, like the Schro¨dinger
operator [18,19], the p−laplacian [3,4], the biharmonic operator [7], p&q−laplacian, fractional
laplacian [24,25], magnetic laplacian [5,6] or quasilinear operators [2,8,9].
Theplanofthepaperisthefollowing. InSection2werecallsomebasicfacts,wepresentsome
preliminaries and the variational setting for the problem. Section 3 is devoted to prove some
compactnessproperties;asabyproductweprovetheexistenceofagroundstatesolutionforour
problem, that is a solution having minimal energy. In Section 4 we introduce the barycenter
map, we show some of its properties and prove, by means of the Ljusternick-Schnirelamnn
Theory, Theorem 1.1.
Notations. In the paper we will denote with |·| the usual Lp norm in RN; we denote with
p
B (x) the closed ball in RN centered in x with radius r > 0, with Bc(x) its complementary;
r r
if x = 0 we simply write B ; moreover the letters C,C ,C ,... will denote generic positive
r 1 2
constants (whose value may change from line to line). Other notations will be introduced
whenever we need.
2. Preliminaries
2.1. Some well known facts. Before to introduce the variational setting of our problem, we
recall some basic facts concerning the fractional Sobolev spaces and their embeddings.
Given β ∈ (0,1), the fractional Laplacian (−∆)β is the pseudodifferential operator which
can be defined via the Fourier transform
F((−∆)βu)= |·|2βFu,
or, if u has sufficient regularity, by
C u(z+y)−u(z−y)−2u(z)
(−∆)βu(z) = − N,β dy, z ∈RN,
2 RN |y|N+2β
Z
where C is a suitable normalization constant.
N,β
4 E.G.MURCIAANDG.SICILIANO
For s ∈ (0,1) let
Hs(RN) = u∈ L2(RN) : (−∆)s/2u∈ L2(RN)
n o
be the Hilbert space with scalar product and (squared) norm given by
(u,v) = (−∆)s/2u(−∆)s/2v+ uv, kuk2 = |(−∆)s/2u|2+|u|2.
2 2
RN RN
Z Z
It is known that Hs(RN) ֒→ Lp(RN),p ∈ [2,2∗] with 2∗ := 2N/(N − 2s). Moreover the
s s
embedding of Hs(Ω) is compact if Ω ⊂ RN is bounded and p 6= 2∗.
s
We will consider also the homogeneous Sobolev spaces H˙α/2(RN) defined as the completion
of C∞(RN) with respect to the norm |(−∆)α/4u| . This is a Hilbert space with scalar product
c 2
and (squared) norm
(u,v) = (−∆)α/4u(−∆)α/4v, kuk2 = |(−∆)α/4u|2.
H˙α/2 H˙α/2 2
RN
Z
It is well known that H˙α/2(RN) ֒→ L2∗α/2(RN),2∗ = 2N/(N−α). For more general facts about
α
the fractional Laplacian we refer the reader to the beautiful paper [22].
We recall here another fact that will be frequently used:
(2.1) ∀ε> 0 ∃M > 0 : f(u)u ≤ ξ u2+M |u|q0+1, ∀u∈ Hs(RN).
ξ ξ
RN RN RN
Z Z Z
This simply follows by (f2) and (f3).
2.2. The variational setting. It is easily seen that, just performing the change of variables
w(x) := u(x/ε),ψ(x) := φ(x/ε), problem (P ) can be rewritten as
ε
(−∆)su+V(εx)u+φ(x)u = f(u)
(P∗)
ε ( (−∆)α/2φ= εα−θγαu2,
to which we will refer from now on.
A usual “reduction” argument can be used to deal with a single equation involving just u.
Indeed for every u ∈ Hs(RN) the second equation in (P∗) is uniquely solved. Actually, for
ε
future reference, we will prove a slightly more general fact.
Let us fix two functions u,w ∈ Hs(RN) and consider the problem
(−∆)α/2φ = εα−θγ uw,
α
(Q )
ε
( φ∈ H˙α/2(RN)
whose weak solution is a function φ˜∈ H˙α/2(RN) such that
∀v ∈ H˙α/2(RN): (−∆)α/4φ˜(−∆)α/4v = εα−θγ uwv.
α
RN RN
Z Z
For every v ∈ H˙α/2(RN), by the H¨older inequality and the continuous embeddings, we have
RN uwv ≤ |u|N4+Nα|w|N4+Nα|v|2∗α/2 ≤ CkukkwkkvkH˙α/2
(cid:12)Z (cid:12)
deducing that the m(cid:12)ap (cid:12)
(cid:12) (cid:12)
T :v ∈ H˙α/2(RN) 7−→ uwv ∈ R
u,w
RN
Z
FRACTIONAL SCHRO¨DINGER-POISSON SYSTEM 5
is linear and continuous: then there exists a unique solution φ ∈ H˙α/2(RN) to (Q ).
ε,u,w ε
Moreover thissolution hastherepresentation bymeansoftheRieszkernelK (x) = γ−1|x|α−N,
α α
hence
1
φ = εα−θ ⋆(uw).
ε,u,w |·|N−α
Furthermore
(2.2) kφ k = εα−θkT k ≤ εα−θCkukkwk
ε,u,w H˙α/2 u,w L(H˙α/2;R)
and then, for ζ,η ∈ Hs(RN)
(2.3) φε,u,wζη ≤ |φε,u,w|2∗ |ζ| 4N |η| 4N ≤ εα−θCekukkwkkζkkηk
RN α/2 N+α N+α
Z
where C is a suitable embedding constant. Altough its value is not important, we will refer
e
to this constant later on.
A particular case of the previous situation is when u = w. In this case we simplify the
notation and write
• T (v) := T (v) = u2v, and
u u,u RN
• φε,u for the unique sRolution of the second equation in (Pε∗) for fixed u∈ Hs(RN). Then
kφ k ≤ εα−θCkuk2
ε,u H˙α/2
and the map
u∈ Hs(RN)7−→ φ ∈ H˙α/2(RN)
ε,u
is bounded.
Observe also that
(2.4) u2n → u2 in LN2+Nα(RN)=⇒ Tun −→ Tu as operators =⇒ φε,un −→ φε,u in H˙α/2(RN).
For convenience let us define the map (well defined by (2.3))
A :u∈ Hs(RN) 7−→ φ u2 ∈ R.
ε,u
RN
Z
Then
(2.5) |A(u)| ≤ εα−θC kuk4
e
(where C is the same constant in (2.3)). Some relevant properties of φ and A are listed
e ε,u
below. Although thesepropertiesareknown tobetrue, wearenotable tofindthem explicitely
in the literature; so we prefer to give a proof here.
Lemma 2.1. The following propositions hold.
(i) For every u∈ Hs(RN) :φ ≥ 0;
ε,u
(ii) for every u ∈ Hs(RN),t ∈ R : φ = t2φ ;
ε,tu ε,u
(iii) if u ⇀ u in Hs(RN) then φ ⇀ φ in H˙α/2(RN);
n ε,un ε,u
(iv) A is of class C2 and for every u,v,w ∈ Hs(RN)
A′(u)[v] = 4 φ uv, A′′(u)[v,w] = 4 φ vw+8 φ uv,
ε,u ε,u ε,u,w
RN RN RN
Z Z Z
(v) if u → u in Lr(RN), with 2 ≤ r < 2∗ , then A(u )→ A(u);
n s n
(vi) if u ⇀ u in Hs(RN) then A(u −u) = A(u )−A(u)+o (1).
n n n n
6 E.G.MURCIAANDG.SICILIANO
Proof. Items (i) and (ii) follow directly by the definition of φ .
ε,u
To prove (iii), let v ∈ C∞(RN); we have
c
(−∆)α/4(φ −φ )(−∆)α/4v = (u2 −u2)v
ε,un ε,u n
RN RN
Z Z
1/2 1/2
≤ |v| (u −u)2 (u +u)2
∞ n n
(cid:16)Zsuppv (cid:17) (cid:16)Zsuppv (cid:17)
→ 0.
The conclusion then follows by density.
The proof of (iv) is straightforward: we refer the reader to [23].
To show (v), recall that 2 < 4N < 2∗. Since by assumption |u2| → |u2| and
N+α s n 2N 2N
N+α N+α
u2 → u2 a.e. in RN, using the Brezis-Lieb Lemma, u2 → u2 in LN2+Nα RN . But then using
n n
(2.4) we get φε,un → φε,u in L2∗α/2(RN). Consequently (cid:0) (cid:1)
|A(u )−A(u)| ≤ |φ u2 −φ u2| ≤ |(φ −φ )u2|+ |φ u2 −u2 |
n ε,un n ε,u ε,un ε,u n ε,u n
RN RN RN
Z Z Z
≤ |φε,un −φε,u|2∗α/2|u2n|N2+Nα +|φε,u|2∗α/2|u2n −u2|N2+Nα (cid:0) (cid:1)
from which we conclude.
To prove (vi), for the sake of simplicity we drop the factor εα−θ in the expression of φ .
ε,u,v
Defining
u2(y)u2(x)
σ := dydx,
RN RN |x−y|N−α
Z Z
u2(y)u2(x) u (y)u(y)u (x)u(x)
σ1 := n dydx, σ2 := n n dydx
n RN RN |x−y|N−α n RN RN |x−y|N−α
Z Z Z Z
u2(y)u (x)u(x) u (y)u(y)u2(x)
σ3 := n n dydx, σ4 := n dydx,
n RN RN |x−y|N−α n RN RN |x−y|N−α
Z Z Z Z
it is easy to check that
A(u −u)−A(u )+A(u) = 2σ+2σ1 +4σ2 −4σ3 −4σ4.
n n n n n n
Now we claim that, whenever u ⇀ u in Hs(RN),
n
lim σi = σ, i= 1,2,3,4
n
n→∞
which readily gives the conclusion.
We prove hereonly the cases i =1,2 since the proof of the other cases is very similar. Recall
that
u2(y) u2(y)
φ (x) = dy, φ (x) = n dy.
ε,u RN |x−y|N−α ε,un RN |x−y|N−α
Z Z
Since u2 ∈ LN2+Nα(RN)= L(2∗α/2)′(RN) and by item (iii) it holds φε,un → φε,u in L2∗α/2(RN), we
conclude that
σ1 = φ u2 → φ u2 = σ
n ε,un ε,u
RN RN
Z Z
and the claim is true for i= 1.
For i= 2 recall that
u (y)u(y)
n
φ (x)= dy.
ε,un,u RN |x−y|N−α
Z
FRACTIONAL SCHRO¨DINGER-POISSON SYSTEM 7
Firstweshowthatφ → φ a.e. inRN. Givenξ > 0andchoosingR >1/ξ, N < p < N
ε,un,u ε,u 2s N−α
and N < q (so that 2p′,2q′ ∈ (2,2∗)), we have, for large n:
N−α s
dy 1/p
|φε,un,u(x)−φε,u(x)| ≤ |un−u|L2p′(BR(x))|u|L2p′(BR(x)) |x−y|p(N−α)
(cid:16)Z|y−x|<R (cid:17)
1/q
dy
+ |un−u|L2q′(BRc(x))|u|L2q′(BRc(x)) Z|y−x|≥R |x−y|q(N−α)!
≤ C ξ+C ξN−α,
1 2
concluding the pointwise convergence. Moreover by the Sobolev embedding and using (2.2),
|φε,un,uun|2 ≤ |φε,un,u|2∗α/2|un|2N/α ≤ C1kunk2kuk ≤ C2
and therefore, up to subsequence, φ u ⇀ φ u in L2(RN), by [26, Lemma 4.8]. Since
ε,un,u n ε,u
u∈ L2(RN)
σ2 = φ u u → φ u2 = σ
n ε,un,u n ε,u
RN RN
Z Z
and the claim is proved for i = 2. (cid:3)
We introduce now the variational setting for our problem. Let us define the Hilbert space
W := u∈ Hs(RN): V(εx)u2 < ∞
ε
(cid:26) ZRN (cid:27)
endowed with scalar product and (squared) norm given by
(u,v) := (−∆)s/2u(−∆)s/2v+ V(εx)uv
ε
RN RN
Z Z
and
kuk2 := |(−∆)s/2u|2+ V(εx)u2.
ε
RN RN
Z Z
Then it is standard to see that the critical points of the C2 functional (see Lemma 2.1 (iv))
1 1 1
I (u) := |(−∆)s/2u|2+ V(εx)u2+ φ u2− F(u),
ε ε,u
2 RN 2 RN 4 RN RN
Z Z Z Z
on W are weak solutions of problem (P∗).
ε ε
By defining
N := u∈ W \{0} : J (u) = 0 ,
ε ε ε
where n o
J (u) := I′(u)[u] = kuk2 + φ u2− f(u)u,
ε ε ε ε,u
RN RN
Z Z
we have, by standard arguments:
Lemma 2.2. For every u ∈ N , J′(u)[u] < 0 and there are positive constants h ,k such that
ε ε ε ε
kuk ≥ h ,I (u) ≥ k . Furthermore, N is diffeomorphic to the set
ε ε ε ε ε
S := {u∈ W :kuk = 1, u > 0 a.e.}.
ε ε ε
N is the Nehari manifold associated to I . By the assumptions on f, the functional I has
ε ε ε
the Mountain Pass geometry. This is standard but we give the easy proof for completeness.
(MP1) I (0) = 0;
ε
8 E.G.MURCIAANDG.SICILIANO
(MP2) since, for every ξ > 0 there exists M > 0 such that F(u) ≤ ξu2+M |u|q0+1, we have
ξ ξ
1
I (u) ≥ kuk2 − F(u)
ε 2 ε RN
Z
1
≥ kuk2 −ξC kuk2 −M C kukq0+1
2 ε 1 ε ξ 2 ε
and we conclude I has a strict local minimum at u = 0;
ε
(MP3) finally, since (f4) implies F(t) ≥ CtK for t > 0, with K > 4 (and less then q +1), fixed
0
v ∈ C∞(RN),v > 0 we have
c
t2 t4
I (tv) = kvk2 + φ v2− F(tv)
ε 2 ε 4 RN ε,v RN
Z Z
t2 t4
≤ kvk2 + φ v2−CtK vK
2 ε 4 RN ε,v RN
Z Z
concluding that the functional is negative for suitable large t.
Then denoting with
c := inf sup I (γ(t)), H = γ ∈ C([0,1],W ) :γ(0) = 0,I (γ(1)) < 0
ε ε ε ε ε
γ∈Hε t∈[0,1]
n o
the Mountain Pass level, and with
m := inf I (u)
ε ε
u∈Nε
the ground state level, it holds, in a standard way, that
(2.6) c = m = inf supI (tu).
ε ε ε
u∈Wε\{0} t≥0
It is known that for “perturbed” problems a major role is played by the problem at infinity
that we now introduce.
2.3. The problem at “infinity”. Let us consider the “limit” problem (the autonomous
problem) associated to (P∗), that is
ε
(−∆)su+µu= f(u)
(A )
µ u∈ Hs(RN)
(cid:26)
where µ > 0 is a constant. The solutions are critical points of the functional
1 µ
E (u) = |(−∆)s/2u|2+ u2− F(u).
µ
2 RN 2 RN RN
Z Z Z
in Hs(RN). Denoting with Hs(RN) simply the space Hs(RN) endowed with the (equivalent
µ
squared) norm
kuk2 := |(−∆)s/2u|2+µ|u|2,
Hs 2 2
µ
by the assumptions of the nonlinearity f, it is easy to see that the functional E has the
µ
Mountain Pass geometry with Mountain Pass level
c∞ := inf sup E (γ(t)), H := γ ∈ C([0,1],Hs(RN)) : γ(0) = 0,E (γ(1)) < 0 .
µ µ µ µ µ
γ∈Hµ t∈[0,1]
n o
Introducing the set
M := u∈ Hs(RN)\{0} :kuk2 = f(u)u
µ Hs
µ RN
n Z o
FRACTIONAL SCHRO¨DINGER-POISSON SYSTEM 9
it is standard to see that
• M has a structure of differentiable manifold (said the Nehari manifold associated to
µ
E ),
µ
• M is bounded away from zero and radially homeomorfic to the unit sphere,
µ
• the mountain pass value c∞ coincide with the ground state level
µ
m∞ := inf E (u) > 0.
µ µ
u∈Mµ
The symbol “∞” in the notations is just to recall we are dealing with the limit problem. In
the sequel we will mainly deal with µ = V and µ = V (whenever this last one is finite). Of
0 ∞
course the inequality
m ≥ m∞
ε V0
holds.
3. Compactness properties for I ,E :
ε µ
existence of a ground state solution
We begin by showing the boundedness of the Palais-Smale sequences for E in Hs(RN) and
µ µ
I in W . Let {u } ⊂ Hs(RN) be a Palais-Smale sequence for E , that is, |E (u )| ≤ C and
ε ε n µ µ µ n
E′(u ) → 0. Then, for large n,
µ n
1 1 1 1
C +ku k > E (u )− E′(u )[u ] = − ku k2 + (f(u )u −KF(u ))
n Hµs µ n K µ n n (cid:18)2 K(cid:19) n Hµs K ZRN n n n
1 1
≥ − ku k2 ,
2 K n Hµs
(cid:18) (cid:19)
and thus {u } is bounded. Similarly we conclude for I , using that
n ε
1 1 1 1 1 1
I (u )− I′(u )[u ]= − ku k2+ − φ u2 + (f(u )u −KF(u ))
ε n K ε n n (cid:18)2 K(cid:19) n ε (cid:18)4 K(cid:19)ZRN ε,un n K ZRN n n n
1 1
≥ − ku k2.
2 K n ε
(cid:18) (cid:19)
In order to prove compactness, some preliminary work is needed. Let us recall the following
Lions type lemma, whose proof can be found in [21, Lemma 2.3].
Lemma 3.1. If {u } is bounded in Hs(RN) and for some R > 0 and 2 ≤ r < 2∗ we have
n s
sup |u |r → 0 as n → ∞,
n
x∈RNZBR(x)
then u → 0 in Lp(RN) for 2 < p < 2∗.
n s
Then we can prove the following
Lemma 3.2. Let {u } ⊂ W be bounded and such that I′(u ) → 0. Then we have either
n ε ε n
a) u → 0 in W , or
n ε
b) there exist a sequence {y } ⊂ RN and constants R,c> 0 such that
n
liminf u2 ≥ c> 0.
n
n→+∞
ZBR(yn)
10 E.G.MURCIAANDG.SICILIANO
Proof. Suppose that b) does not occur. Using Lemma 3.1 it follows
u → 0 in Lp(RN) for p ∈ (2,2∗).
n s
Using (2.1), the boundedness of {u } in L2(RN) and the fact that u → 0 in Lq0+1(RN), we
n n
conclude that
f(u )u → 0.
n n
RN
Z
Finally, since
ku k2− f(u )u ≤ku k2+ φ u2 − f(u )u = I′(u )[u ]= o (1),
n ε n n n ε ε,un n n n ε n n n
RN RN RN
Z Z Z
it follows that u → 0 in W . (cid:3)
n ε
Intherestofthepaperweassume,withoutlossofgenerality, that0 ∈ M,thatis,V(0) = V .
0
Lemma 3.3. Assume that V < ∞ and let {v } ⊂ W be a (PS) sequence for I such that
∞ n ε d ε
v ⇀ 0 in W . Then
n ε
v 6→ 0 in W =⇒ d≥ m∞ .
n ε V∞
Proof. Observe, preliminarly, that by condition (V1) it follows that
(3.1) ∀ξ > 0 ∃R = Rξ > 0: V(εx) > V∞−ξ, ∀x ∈/ BRe.
Let {t } ⊂(0,+∞) be such that {t v } ⊂ M . We start by showing the following
n n n V∞
e e
Claim: The sequence {t } satisfies limsup t ≤ 1.
n n→∞ n
Supposing by contradiction that the claim does not hold, there exists δ > 0 and a subsequence
still denoted by {t }, such that
n
(3.2) t ≥ 1+δ for all n ∈ N.
n
Since {v } is a bounded (PS) sequence for I , I′(v )[v ] = o (1), that is,
n d ε ε n n n
kv k2+ φ v2 = f(v )v +o (1).
n ε ε,vn n n n n
Rn Rn
Z Z
Moreover, since {t v } ⊂M , we get
n n V∞
kt v k2 = f(t v )t v .
n n Hs n n n n
V∞ RN
Z
These equalities imply that
f(t v )
n n −f(v ) v = (V −V(εx))v2 − φ v2 +o (1),
ZRN (cid:18) tn n (cid:19) n ZRN ∞ n ZRN ε,vn n n
and thus
f(t v )
(3.3) n n −f(v ) v ≤ (V −V(εx))v2 +o (1).
RN tn n n RN ∞ n n
Z (cid:16) (cid:17) Z
Using (3.1), the fact that vn → 0 in L2(BRe) and that {vn} is bounded in Wε, let us say by
some constant C > 0, we deduce by (3.3)
f(t v )
n n
(3.4) ∀ξ > 0 : −f(v ) v ≤ ξC +o (1).
n n n
ZRN (cid:18) tn (cid:19)
Since v 6→ 0 in W , we may invoke Lemma 3.2 to obtain {y } ⊂ RN and R,c > 0 such that
n ε n
(3.5) v2 ≥ c.
n
ZBR(yn)