Ground states for superlinear fractional Schr\"odinger equations in \R^{N}

0.2 MB·

by  Vincenzo Ambrosio

GROUND STATES FOR SUPERLINEAR FRACTIONAL
SCHRO¨DINGER EQUATIONS IN RN
6
1 VINCENZO AMBROSIO
0
2

n
a
J Abstract. Inthispaperwestudygroundstatesofthefollowingfractional

3 Schr¨odinger equation
2 ( ∆)su+V(x)u=f(x,u) in RN,
] (cid:26) u− Hs(RN)
P ∈
wheres (0,1),N >2sandf isacontinuousfunctionsatisfyingasuitable
A ∈
growth assumption weaker than the Ambrosetti-Rabinowitz condition.
.
h WeconsiderthecaseswhenthepotentialV(x)is1-periodicorhasabounded
t potential well.
a
m
[
1. Introduction

1
v Recently there has been an increasing interest in the study of nonlinear partial
4 differential equations driven by fractional operators, from a pure mathemati-
8
cal point of view as well as from concrete applications, since these operators
2
6 naturally arise in several fields of research like obstacle problem, phase transi-
0
tion, conservation laws, financial market, flame propagations, ultra relativistic
.
1
limits of quantum mechanic, minimal surfaces and water wave. The literature
0
6 is too wide to attempt a reasonable list of references here, so we derive the
1 reader to the work by Di Nezza, Patalluci and Valdinoci [7], where a more
:
v extensive bibliography and an introduction to the subject are given.
i
X The present paper is devoted to the study of the following equation:
ar ( ∆)su+V(x)u = f(x,u) in RN,
(cid:26) u− Hs(RN) (1.1)
∈
wheres (0,1),N > 2s, thepotentialV(x)andthenonlinearityf : RN R
R satisfy∈the following assumptions: × →
(V1) V C(RN) and α V(x) β;
(f1) f ∈C(RN R) is 1-≤periodic≤in x and
∈ ×
f(x,t)
lim = 0 uniformly in x RN
|t|→∞ t 2∗s−1 ∈
| |
where 2∗ = 2N ;
s N−2s
(f2) f(x,t) = o(t) as t 0 uniformly in x RN.
| | → ∈
1
2 VINCENZOAMBROSIO
Here ( ∆)s can be defined, for smooth functions u, by
−
u(x) u(y)
( ∆)su(x) = c P.V. − dy,
− N,s Z x y N+2s
RN
| − |
where P.V. stands for the Cauchy principal value and c is a normalization
N,s
constant; see [2, 7].
Equation (1.1) arises in the study of the Fractional Schro¨dinger equation
∂ψ
ı +( ∆)sψ = H(x,ψ) in RN R
∂t − ×
when the wave function ψ is a standing wave, that is ψ(x,t) = u(x)e−ıct, where
c is a constant. This equation was introduced by Laskin [13, 14] and comes
from an extension of the Feynman path integral from the Brownian-like to the
Levy-like quantum mechanical paths.
In recent years great attention has been focused on the fractional Schro¨dinger
equation. Felmer, Quaas & Tan [9] studied the existence and regularity of
positive solution to (1.1) with V(x) = 1 for general s (0,1) when f has
∈
subcritical growth and satisfies the Ambrosetti-Rabinowitz condition. Sec-
chi [16, 17] proved some existence results for (1.1) under the assumptions
that the nonlinearity is either of perturbative type or satisfies the Ambrosetti-
Rabinowitz condition. Cheng [6] proved the existence of bound state solutions
for (1.1) in which the potential V(x) is unbounded and f(x,u) = u p−1u with
| |
1 < p < 4s +1.
N
When s = 1, formally, equation in (1.1) reduces to the classical Nonlinear
Schro¨dinger Equation
∆u+V(x)u = f(x,u) in RN, (1.2)
−
which has been extensively studied in the last twenty years and we do not even
try to review the huge bibliography.
To deal with (1.2) many authors supposed that the nonlinear term satisfied
the following condition due to Ambrosetti and Rabinowitz [1]
µ > 2, R > 0 : 0 < µF(x,t) f(x,t)t t R, (AR)
∃ ≤ ∀| | ≥
where F is the primitive of f with respect to the second variable.
This condition is very useful in critical point theory since it ensures the bound-
edness of the Palais-Smale sequences of the functional associated to (1.2).
However, there are many functions which are superlinear at infinity, but do
not satisfy (AR). At this purpose, we would note that from the condition (AR)
and the fact that µ > 2, it follows that
F(x,t) t
(f3) lim = + , where F(x,t) = f(x,τ)dτ.
|t|→∞ |t|2 ∞ Z0
Ofcourse, alsocondition(f3)characterizes thenonlinearityf tobesuperlinear
at infinity. It is easily seen that the function f(x,t) = tlog(1+ t ) verifies (f3)
| |
GROUND STATES FOR SUPERLINEAR FRACTIONAL SCHRO¨DINGER EQUATIONS 3
anddoes notsatisfy (AR).Inorder tostudy thenonlinear problem(1.2)andto
drop the condition (AR), Jeanjean in [11] introduced the following assumption
on f:
(f4) There exists λ 1 such that
≥
G(x,θt) λG(x,t) for (x,t) RN R and θ [0,1],
≤ ∈ × ∈
where G(x,t) = f(x,t)t 2F(x,t).
−
The aim of this paper is to investigate solutions of the corresponding fractional
case of problem (1.2) without assuming (AR). Since u = 0 is a trivial solution
to (1.1) by (f2), we will look for nontrivial solutions to (1.1).
Our first result can be stated as follows
Theorem 1. Assume that f satisfies (f1) (f4) and V satisfies (V1) and
−
(V2) V(x) is 1-periodic.
Then there exists a nontrivial ground state solution u Hs(RN) to (1.1).
∈
Oneofthemaindifficulty instudying (1.1)isthenonlocalcharacter ofthefrac-
tional Laplacian ( ∆)s with s (0,1). To overcome this difficulty, Caffarelli
− ∈
and Silvestre [4] showed that it is possible to realize ( ∆)s as an operator
−
that maps a Dirichlet boundary condition to a Neumann boundary condition
via an extension degenerate elliptic problem in RN+1. However, although this
+
approach is very common nowadays (see [2, 3, 10, 18]), in this paper we prefer
to investigate (1.1) directly in Hs(RN) in order to apply the techniques used
to study the case s = 1.
More precisely, we will look for the critical points for the following functional
1 u(x) u(y) 2
(u) = | − | dxdy + V(x)u2(x)dx F(x,u)dx.
J 2hZZR2N x y N+2s ZRN i−ZRN
| − |
By assumptions on f follow easily that has a Mountain Pass geometry.
J
Namely setting
Γ = γ C([0,1],Hs(RN)) : γ(0) = 0 and (γ(1)) < 0
{ ∈ J }
we have Γ = and
6 ∅
c = inf max (γ(t)).
γ∈Γt∈[0,1]J
The value c is called the Mountain Pass level for . Ekeland’s principle [8]
J
guarantees the existence of a Cerami sequence at the level c. Hence, by using
similar arguments to those developed in [12, 15] and the ZN-invariant of the
problem (1.1), we will prove that every Cerami sequence for is bounded and
J
that there exists a subsequence which converges to a critical point for .
J
Finally, we will also consider the potential well case. We will assume that V(x)
satisfies, in addition to (V1), the following condition
(V3) V(x) < V := lim V(y) < , x RN
∞
|y|→∞ ∞ ∀ ∈
4 VINCENZOAMBROSIO
and that f(x,u) = b(x)f(u) where b C(RN) and
∈
¯
0 < b := lim b(y) b(x) b < (1.3)
∞
|y|→∞ ≤ ≤ ∞
for any x RN and f satisfies (f1) (f4).
∈ −
Therefore our problem becomes
( ∆)su+V(x)u = b(x)f(u) in RN
(cid:26) u− Hs(RN) . (1.4)
∈
To study (1.4), we will use the energy comparison method in [12]. More pre-
cisely, introducing the energy functional at infinity
1 u(x) u(y) 2
(u) = | − | dxdy + V u2(x)dx b F(u)dx
J∞ 2 ZZ x y N+2s Z ∞ −Z ∞
h R2N RN i RN
| − |
wewill showthat, under theaboveassumptions onf andV, hasa nontrivial
J
critical point provided that
c < m (1.5)
∞
where
m = inf (u) : u = 0 and ′ (u) = 0 .
∞ {J∞ 6 J∞ }
To prove (1.5) we will exploit that our problem at infinity is autonomous
( ∆)su = V u+b f(u) in RN,
∞ ∞
− −
so it admits a least energy solution satisfying the Pohozaev identity; see [5].
This information will be useful to deduce the existence of a path γ Γ such
∈
that max (γ(t)) < m . Combining these facts, we will be able to prove our
∞
t∈[0,1]J
main second result:
Theorem 2. Let N > 2s. Assume that V satisfies (V1) and (V3), and that
f verifies the assumptions (f1) (f4). Then (1.4) has a ground state.
−
The paper is organized as follows: In Section 2 we introduce a variational
setting of our problem and collect some preliminary results; in Section 3 we
prove the existence of a nontrivial ground state to (1.1) when the potential
V is assumed 1-periodic; finally, under the assumption that V has a bounded
potential well, we verify that it is possible to find a ground state to (1.4).
2. Preliminaries and functional setting
In this preliminary Section, for the reader’s convenience, we collect some basic
results that will be used in the forthcoming Sections.
GROUND STATES FOR SUPERLINEAR FRACTIONAL SCHRO¨DINGER EQUATIONS 5
Let us denote by -the Lq norm of a function u : RN R. We
Lq(RN)
| · | →
define the homogeneous fractional Sobolev space s(RN) as the completion of
C∞(RN) with respect to the norm D
c
u(x) u(y) 2
u 2 := | − | dxdy = [u]2 .
|| ||Ds(RN) ZZR2N x y N+2s Hs(RN)
| − |
We denote by Hs(RN) the standard fractional Sobolev space, defined as the set
of u s(RN) satisfying u L2(RN) with the norm
∈ D ∈
u(x) u(y) 2 1
||u||Hs(RN) := (cid:16)ZZR2N | x −y N+2s| dxdy +ZRN u2dx(cid:17)2
| − |
= [u]2 + u 2 . (2.1)
Hs(RN) | |L2(RN)
For any u Hs(RN), it holds the following Sobolev inequality
∈
|u|L2∗s(RN) ≤ C||u||2Ds(RN).
Now, we recall the following lemmas which will be useful in the sequel.
Lemma 1. [7] Hs(RN) is continuously embedded in Lq(RN) for any q [2,2∗]
and compactly embedded in Lq (RN) for any q [2,2∗). ∈ s
loc ∈ s
Lemma 2. [9] Let N > 2s. Assume that u is bounded in Hs(RN) and it
k
{ }
satisfies
lim sup u (x) 2dx = 0,
k
k→+∞ξ∈RN ZBR(ξ)| |
where R > 0. Then u 0 in Lq(RN) for 2 < q < 2∗.
k → s
At this point, we give the definition of weak solution for the equation
( ∆)su+V(x)u = g in RN. (2.2)
−
Definition 1. Given g L2(RN), we say that u Hs(RN) is a weak solution
∈ ∈
to (2.2) if u satisfies
(u(x) u(y))
− (v(x) v(y))dxdy+ V(x)uvdx = gvdx
ZZR2N x y N+2s − ZRN ZRN
| − |
for all v Hs(RN).
∈
To study solutions to (1.1), we consider the following functional on Hs(RN)
defined by setting
1
(u) = [u]2 + V(x)u2(x)dx F(x,u)dx.
J 2(cid:16) Hs(RN) ZRN (cid:17)−ZRN
By (V1) follows that
[u]2 + V(x) u(x) 2dx
Hs(RN) Z | |
RN
6 VINCENZOAMBROSIO
is a norm which is equivalent to the standard norm defined in (2.1). For such
reason, we will always write
1
(u) = u 2 F(x,u)dx.
J 2|| || −ZRN
In particular, by assumptions on f, we deduce that C1(Hs(RN),R).
J ∈
Let us observe that possesses a Mountain Pass geometry. More precisely,
J
we have the following result, whose simple proof is omitted.
Lemma 3. Under the assumptions (f1) (f4), there exist r > 0 and v
0
Hs(RN) such that v > r and − ∈
0
|| ||
b := inf (u) > (0) = 0 (v ). (2.3)
0
||u||=rJ J ≥ J
In particular
′(u),u = u 2+o( u 2) as u 0,
hJ i || || || || || || →
1
(u) = u 2+o( u 2) as u 0
J 2|| || || || || || →
and, as a consequence
(i) there exists η > 0 such that if v is a critical point for , then v η;
J || || ≥
(ii) for any c > 0 there exists η > 0 such that if (v ) c then v η .
c n n c
J → || || ≥
Therefore, by Lemma 3, follows that
Γ = γ C([0,1],Hs(RN)) : γ(0) = 0 and (γ(1)) < 0 =
{ ∈ J } 6 ∅
and we can define the Mountain Pass level
c = inf max (γ(t)). (2.4)
γ∈Γt∈[0,1]J
Let us point out that, by (2.3), c is positive. Then, by using the Ekeland’s
principle [8], we know that there exists a Cerami sequence v at the level c
n
{ }
for , that is
J
(v ) c and (1+ v ) ′(v ) 0.
n n n ∗
J → || || ||J || →
We conclude this section proving that the primitive F(x,t) of f(x,t) is non-
negative.
Lemma 4. Let us assume that f satisfies (f1),(f2) and (f4). Then F 0 in
RN R. ≥
×
Proof. Firstly we observe that by (f4) follows
G(x,t) = f(x,t)t 2F(x,t) 0 for all (x,t) RN R.
− ≥ ∈ ×
GROUND STATES FOR SUPERLINEAR FRACTIONAL SCHRO¨DINGER EQUATIONS 7
F(x,t)
Fix t > 0. For x RN let us compute the derivative of with respect
∈ t2
to t:
∂ F(x,t) f(x,t)t2 2tF(x,t)
= − 0. (2.5)
∂t t2 t4 ≥
(cid:16) (cid:17)
Moreover by (f2) we get
F(x,t)
lim = 0. (2.6)
t→0+ t2
Puttingtogether(2.5)and(2.6)wededucethatF(x,t) 0forall(x,t) RN
[0,+ ). Analogously, we obtain that F(x,t) 0 for al≥l (x,t) RN (∈ ,0×].
∞ ≥ ∈ × −∞ (cid:3)
3. Existence of ground states to (1.1)
In this Section we give the proof of the Theorem 1. We start proving the
following Lemma, inspired by [12, 15], which guarantees the boundedness of
Cerami sequences for the functional .
J
Lemma 5. Assume that (V1), (f1), (f2), (f3) and (f4) hold true. Let c R.
∈
Then any Cerami sequence for is bounded.
J
Proof. Let v be a Cerami sequence for .
n
{ } J
Assumebycontradictionthat v isunbounded. Thengoingtoasubsequence
n
{ }
we may assume that
(v ) c, v , ′(v ) v 0. (3.1)
n n n ∗ n
J → || || → ∞ ||J || || || →
v
Now we define set w = n . Clearly w is bounded in Hs(RN) and has
n n
v
n
|| ||
unitary norm. We claim to prove that w vanishes, i.e. it holds
n
{ }
lim sup w 2dx = 0. (3.2)
n
n→∞z∈RN ZB2(z)| |
If (3.2) does not hold, there exists δ > 0 such that
sup w 2dx δ > 0.
n
z∈RN ZB2(z)| | ≥
As a consequence, we can choose z RN such that
n
{ } ⊂
δ
w 2dx .
n
Z | | ≥ 2
B2(zn)
Since the number of points in ZN (z ) is less than 4N, then there exists
2 n
ξ ZN (z ) such that ∩ B
n 2 n
∈ ∩B
w 2dx K > 0, (3.3)
n
Z | | ≥
B2(ξn)
8 VINCENZOAMBROSIO
where K := δ2−(2N+1). Now we set w˜ = w ( +ξ ). By using (V1) and that
n n n
·
w has unitary norm, we deduce
n
w˜ 2 = [w˜ ]2 + V(x) w˜ 2dx
|| n|| n Hs(RN) Z | n|
RN
[w˜ ]2 +β V(x) w˜ (x) 2dx
≤ n Hs(RN) Z | n |
RN
= [w ]2 +β w (x) 2dx
n Hs(RN) ZRN | n |
β
[w ]2 +α w (x) 2dx
≤ α(cid:16) n Hs(RN) ZRN | n | (cid:17)
β
[w ]2 + V(x) w 2dx
≤ α(cid:16) n Hs(RN) ZRN | n| (cid:17)
β
= ,
α
that is w˜ is bounded. By Lemma 1, we may assume, going if necessary to a
n
subsequence, that
w˜ w˜ in L2 (RN),
n → loc (3.4)
w˜ (x) w˜(x) a.e. x RN.
n
→ ∈
Then, by (3.3) and (3.4) we get
w˜ 2dx = lim w˜ 2dx = lim w 2dx K > 0, (3.5)
n n
Z | | n→∞Z | | n→∞Z | | ≥
B2(0) B2(0) B2(ξn)
which implies w˜ = 0.
Let v˜ = v w˜ 6. Since w˜ = 0 the set A := x RN : w˜ = 0 has positive
n n n
|| || 6 { ∈ 6 }
Lebesgue measure and v˜ (x) + . In particular, by (f3) we get
n
| | → ∞
F(x,v˜ (x))
n w˜ (x) 2 + . (3.6)
n
v˜ (x) 2 | | → ∞
n
| |
Let us observe that f(x,t) is 1-periodic with respect to x, so
F(x,v )dx = F(x,v˜ )dx. (3.7)
n n
Z Z
RN RN
GROUND STATES FOR SUPERLINEAR FRACTIONAL SCHRO¨DINGER EQUATIONS 9
By (3.1), (3.6), (3.7) and Lemma 4 follow easily that
1 c+o(1) F(x,v )
n
= dx
2 − v 2 Z v 2
|| n||e RN || n||e
F(x,v˜ )
n
= dx
Z v 2
RN || n||e
F(x,v˜ )
n w˜ 2dx (3.8)
n
≥ Z v˜ 2 | | → ∞
A n
| |
which gives a contradiction. Therefore (3.2) holds true. In particular, by
Lemma 2, we get
w 0 in Lq(RN) q (2,2∗).
n → ∀ ∈ s
Now, let ρ > 0 be a real number. By (f1) (f3) and Lemma 4 follow that for
−
any ε > 0 there exists C > 0 such that
ε
0 F(x,ρt) ε( t 2 + t 2∗s)+C t q. (3.9)
ε
≤ ≤ | | | | | |
Since w = 1, by Sobolev inequality we have that there exists c˜ > 0 such
n
|| ||
that
w 2 + w 2∗s c˜. (3.10)
| n|L2(RN) | n|L2∗s(RN) ≤
Taking into account (3.9) and (3.10) we have
limsup F(x,ρw )dx limsup[ε( w 2 + w 2∗s )+C ( w q ]
n→∞ ZRN n ≤ n→∞ | n|L2(RN) | n|L2∗s(RN) ε | n|Lq(RN)
εc˜
≤
and by the arbitrariness of ε we get
lim F(x,ρw )dx = 0. (3.11)
n
n→∞ZRN
Now, let t [0,1] be a sequence such that
n
{ } ⊂
(t v ) := max (tv ). (3.12)
n n n
J t∈[0,1]J
By using (3.1) we can see that 2 j v −1 (0,1) for n sufficiently large and
n
j N. Taking ρ = 2√j in (3.11)p, we||ob|t|ain∈
∈
(t v ) (2 jw )
n n n
J ≥ J
p
= 2j F(x,2 jw )dx j
n
−Z ≥
RN p
for n large enough and for all j N. Then
∈
(t v ) + . (3.13)
n n
J → ∞
10 VINCENZOAMBROSIO
Since (0) = 0 and (v ) c we deduce that t (0,1). By (3.12) we get
n n
J J → ∈
d
′(t v ),t v = t (tv ) = 0. (3.14)
n n n n n n
hJ i dtJ (cid:12)t=tn
(cid:12)
Indeed, putting together (3.1), (3.14) and (f4), w(cid:12)e can see
2 1
(t v ) = 2 (t v ) ′(t v ),t v
n n n n n n n n
λJ λ J −hJ i
(cid:16) (cid:17)
1
= f(x,t v )t v 2F(x,t v ) dx
n n n n n n
λ Z −
RN(cid:16) (cid:17)
1
= G(x,t v )dx
n n
λ Z
RN
G(x,t v )dx
n n
≤ Z
RN
= f(x,v )v 2F(x,v ) dx
n n n
Z −
RN(cid:16) (cid:17)
= 2 (v ) ′(v ),v 2c
n n n
J −hJ i →
which is incompatible with (3.13). Thus v is bounded.
n
{ } (cid:3)
Remark 1. Let us observe that the conclusion of Lemma 5 holds true if we
consider f(x,t) = b(x)f(t) with b C(RN) and 0 < b b(x) b < for
0 1
anyx RN. In fact, inthis case, th∈econtradiction in(3.8≤) follow≤sby rep∞lacing
∈
(3.7) by
b
0
b(x)F(v )dx b(x)F(v˜ )dx.
n n
ZRN ≥ b1 ZRN
Now we prove that, up to a subsequence, our bounded Cerami sequence u
n
{ }
converges weakly to a non-trivial critical point for .
J
Proof of Theorem 1. Let c be the Mountain Pass level defined in (2.4). We
know that c > 0 and that there exists a Cerami sequence u for , which is
n
bounded in Hs(RN) by Lemma 5. { } J
We define
δ := lim sup u 2dx.
n
n→∞z∈RN ZB2(z)| |
If δ = 0, then by Lemma 2 we have that u 0 in Lq(RN) for all q (2,2∗).
n → ∈ s
Analogously to (3.11) we can see
lim F(x,u )dx = 0,
n
n→∞ZRN
lim f(x,u )u dx = 0.
n n
n→∞ZRN