Ground state solutions for a nonlinear Choquard equation

0.14 MB·

by  Luca Battaglia

Ground state solutions for a nonlinear Choquard
equation
∗
Luca Battaglia
7
1
0
2
Abstract

n
a
J WediscusstheexistenceofgroundstatesolutionsfortheChoquardequa-

9 tion
−∆u+u=(Iα∗F(u))F′(u) in RN.
]
P Weprovetheexistence of solutions undergeneral hypotheses, investigat-
|u|p
A ing in particular the case of a homogeneous nonlinearity F(u) = .
p
h. The cases N = 2 and N ≥ 3 are treated differently in some steps. The
t solutions are found through a variational mountain-pass strategy.
a
Theresults presented are contained in the papers [8, 2].
m
[

1
v 1 Introduction
6
7
3
We investigate the existence of solutions for nonlinear Choquard equations of
2
0 the form
. −∆u+u=(I ∗F(u))F′(u) in RN, (1.1)
1 α
0 where ∆ is the standard Euclidean laplacian, ∗ indicates the convolution, F ∈
7 C1(R,R)isasmoothnonlinearityandI :RN →Ris,forα∈(0,N),the Riesz
1 α
: potential:
v Γ N−α 1
i I (x):= 2 . (1.2)
X α Γ α2(cid:0) πN2(cid:1)2α|x|N−α
r
a Problem (1.1) can be seen as a non(cid:0)-lo(cid:1)cal counterpart of the very well-known
scalar field equation
−∆u+u=G′(u) in RN, (1.3)
which can be formally recovered from (1.1) by letting α go to 0 and setting
F2
G= .
2
Problem (1.3) has been widely studied since many years. General existence
results were provided in [4] when N ≥ 3 and [3] (when N = 2) under mild
∗
SapienzaUniversit`adiRoma,DipartimentodiMatematica,PiazzaleAldoMoro5,00185
[email protected]
1
hypotheses on G.
Anyway, the argument from both [4] and [3] does not seem to be suitable to
attack problem (1.3): roughly speaking, the authors use a constrained mini-
mization technique and then a dilation to get rid of the Lagrangianmultiplier,
which does not work in our case because of the scaling properties of the Riesz
potential (1.2).
We study the problem(1.1) variationally: its solutions are criticalpoints of the
following energy functional on H1 RN :
1 (cid:0) (cid:1) 1
I(u)= |∇u|2+|u|2 − (I ∗F(u))F′(u). (1.4)
2ˆRN 2ˆRN α
(cid:0) (cid:1)
In particular, we look for solutions at a mountain-pass level b defined by
b:= inf sup I(γ(t)), (1.5)
γ∈Γt∈[0,1]
with
Γ:= γ ∈C [0,1],H1 RN ; γ(0)=0, I(γ(1))<0 .
In particular, we(cid:8)by-pass(cid:0)the issue(cid:0)of P(cid:1)a(cid:1)lais-Smale sequences by(cid:9)a scaling trick
introduced in [5], which basically allows us to consider Palais-Smale sequences
also asymptotically satisfying the Pohoˇzaevidentity
N −2 N N +α
P(u):= |∇u|2+ |u|2− (I ∗F(u))F(u)=0, (1.6)
2 ˆRN 2 2 ˆRN α
for which convergence is easier to be proved.
We canshowexistenceofsolutionsundergeneralhypotheses,inthe samespirit
of [4, 3]. In the particular yet very important case of a power-typenonlinearity
|u|p α N +α
F(u) = such hypotheses are equivalent to 1+ < p < , which in
p N N −2
[7] is shown to be also a necessary condition. This shows that the hypotheses
we make are somehow natural.
Wealsoshowthatthemountain-typepasssolutionisalsoagroundstate,namely
an energy-minimizing solutions: it satisfies
I(u)=c:=inf I(v): v ∈H1 RN \{0} solves (1.1) . (1.7)
We first show the existenc(cid:8)e of mountain-p(cid:0)ass(cid:1)solutions in Sectio(cid:9)n 2 and then
in Section 3 we prove that they are actually ground states. Such results were
originallypresentedin[8]forthedimensionN ≥3andin[2]forthecaseN =2.
2 Existence of mountain-pass solutions
We show here existence of a solution for (1.1) under general hypotheses on F.
First of all, we want to exclude the trivial case of an identically vanishing F:
(F ) There exists s ∈R such that F(s )6=0.
0 0 0
Then, we also need some growth assumptions which give a well-posed varia-
tionalformulation,namelyaenergyfunctionalI beingwell-definedonH1 RN .
(cid:0) (cid:1)
2
Such assumptions are different depending whether the dimension is two or it is
greater, since the limiting-case embeddings in Sobolev spaces are different: in
the higher-dimensional case, we impose a power-type growth whereas in R2 we
require one of exponential type:
(N ≥3) (F1) There exists C >0 such that |F′(s)|≤C |s|Nα +|s|Nα+−22 for any s>0
(cid:16) (cid:17)
(N =2) (F1′) Foranyθ >0thereexistsCθ >0suchthat|F′(s)|≤Cθmin 1,|s|α2 eθ|s|2
for any s>0.
(cid:8) (cid:9)
It is not hard to see that (F ), combined with Sobolev and Hardy-Littlewood-
1
Sobolevinequality,impliesthefinitenessofthe term (I ∗F(u))F(u),hence
ˆ α
RN
the well-posedness and smoothness of the functional I defined by (1.4). In
dimension two we need, in place of Sobolev’s inequality, a special form of the
Moser-Trudinger inequality on the whole plane, which was given in [1]:
∀β ∈(0,4π)∃C >0 such that |∇u|2 ≤1 ⇒ min 1,u2 eβu2 ≤C |u|2
β ˆ ˆ βˆ
R2 R2 R2
(cid:8) (cid:9) (2.1)
Thelasthypothesesweneedisasortofsub-criticality withrespecttothecritical
power in Hardy-Littlewood-Sobolev inequality. Again, we state the condition
differently depending onthe dimension, since in dimension2 there is no critical
Sobolev exponent:
F(s) F(s)
(N ≥3) (F ) lim = lim =0
2 s→0|s|1+Nα s→+∞|s|NN+−α2
F(s)
(N =2) (F′) lim =0
2 s→0|s|1+Nα
Precisely, the result we present is the following:
Theorem 2.1.
Assume F satisfies (F ),(F ),(F ) if N ≥ 3 and (F ),(F′),(F′) if N = 2.
0 1 2 0 1 2
Then, the problem (1.1) has a non-trivial solution u∈H1 RN \{0}.
(cid:0) (cid:1)
We start by showing the existence of a Pohoˇzaev-Palais-Smale sequence. We
argue as in [5] to get the asymptotical Pohoˇzaevidentity.
Lemma 2.2.
AssumeF satisfied(F ),(F )(or, in caseN =2,(F ),(F′)). Then, thereexists
0 1 0 1
a sequence (un)n∈N in H1 RN such that:
I(u ) → b I(cid:0)′(u )(cid:1) → 0 in H1 RN ′ P(u ) → 0
n n n
n→+∞ n→+∞ n→+∞
(cid:0) (cid:1)
Proof.
We divide the proof in three steps: first we show that the mountain-pass level
(1.5) is not degenerate and then we apply a variant of the mountain-pass prin-
ciple.
3
Step 1: b>0 We suffice to show that Γ 6= ∅, namely that there exists some u ∈
0
H1 RN with I(u )<0.
0
By (F ), we can choose s such that F(s ) 6= 0, therefore if we take a
0 0 0
(cid:0) (cid:1)
smooth v approximating s 1 we easily get (I ∗F(v ))F(v ) > 0.
0 0 B1 ˆ α 0 0
RN
·
If now we consider v =v , we get
t 0 t
(cid:16) (cid:17)
tN−2 tN tN+α
I(v )= |∇v |2+ |v |2− (I ∗F(v ))F(v ),
t 2 ˆRN 0 2 ˆRN 0 2 ˆRN α 0 0
(2.2)
which is negative for large t, so we can take u =v with t≫1.
0 t
Step 2: b<+∞ We need to show that for any γ ∈ Γ there exists t such that I(γ(t )) ≥
γ γ
ε>0.
If |∇u|2+|u|2 ≤ δ ≪ 1, then by assumption (F ) and H-L-S and
ˆ 2
RN
Sobolev(cid:0)’s inequality(cid:1)we get
NN+−α2 1+Nα 1
(I ∗F(u))F(u)≤C |∇u|2 + |u|2 ≤ |∇u|2+|u|2 ,
ˆRN α (cid:18)ˆRN (cid:19) (cid:18)ˆRN (cid:19) ! 4ˆRN
(cid:0) (cid:1)
1
which means I(u) ≥ |∇u|2+|u|2 , and the same can be proved
4ˆRN
similarly when N =2. (cid:0) (cid:1)
Now,foranyfixedγ ∈Γwecantaket suchthat |∇γ(t )|2+|γ(t )|2 =
γ ˆ γ γ
R2
δ (cid:0) (cid:1)
δ and we get I(γ(t ))≥ =:ε.
γ
4
Step 3: Conclusion We apply consider the functional I :R×H1 RN →R defined by
I(σ,v):=I v e−σ· = e(N−2)σ e |∇v|2+(cid:0)eNσ(cid:1) |v|2−e(N+α)σ (I ∗F(v))F(v).
2 ˆRN 2 ˆRN 2 ˆRN α
(cid:0) (cid:0) (cid:1)(cid:1)
e
Byapplying toI the standardmin-max principle (see [9]for instance)we
′
getasequence(σn,vn)n∈N withI(σn,vn) → bandI(σn,vn) → 0,
e n→+∞ n→+∞
which is equivalent to what the Lemma required.
e e
ToproveTheorem2.1weneedtoshowtheconvergenceofthePohoˇzaev-Palais-
Smale sequence we just found. Here we need the sub-criticality assumption
(F ),(F′)
2 2
Lemma 2.3.
Assume F satisfied (F1),(F2) (or, in case N = 2, (F1′),(F2′)) and (un)n∈N
satisfies
I(u ) is bounded I′(u ) → 0 in H1 RN ′ P(u ) → 0.
n n n
n→+∞ n→+∞
(cid:0) (cid:1)
Then, up to subsequences,
4
• either u → 0 strongly in H1 RN
n
n→+∞
(cid:0) (cid:1)
• or un(·−xn) ⇀ u weakly for some (xn)n∈N in RN and u∈H1 RN \
n→+∞
{0}. (cid:0) (cid:1)
Proof.
Assume the firstalternativedoesnotoccur. Then, weshowitweaklyconverges
to some u6≡0.
Step 1: (un)n∈N is bounded It follows by just writing
α+2 α P(u )
|∇u |2+ |u |2 =I(u )− n → b
2(N +α)ˆRN n 2(N +α)ˆRN n n N +α n→+∞
1
Step 2: sup |u |p ≥ By using the asymptotic Pohoˇzaev identity it is not hard to see that
ˆ n C
x∈RN B1(x)
inf (I ∗F(u ))F(u )>0. Moreover,(F ) implies, for any ǫ>0,p∈
n ˆRN α n n 2
2N
2, ,
N −2
(cid:18) (cid:19)
|F(s)|N2+Nα ≤ε |s|2+|s|N2−N2 +Cε|s|p,
(cid:16) (cid:17)
therefore, by the following inequality from [6]
1−2
p
|u |p ≤C |∇u |2+|u |2 sup |u |p ,
ˆ n ˆ n n ˆ n
RN (cid:18) RN (cid:19) x∈RN B1(x) !
(cid:0) (cid:1)
we get
1−2
sup |u |p p ≥ 1 RN |un|p
x∈RNˆB1(x) n ! C RN (|´∇un|2+|un|2)
´
1
≥ C ˆ |F(un)|N2+Nα −εˆ |u|2+|u|N2−N2
ε (cid:18) RN RN (cid:16) (cid:17)(cid:19)
N
1 N+α 1
≥ (I ∗F(u ))F(u ) −Cε |∇u |2+|u |2 ≥ .
C′ ˆ α n n ˆ n n C
ε (cid:18) RN (cid:19) RN !
(cid:0) (cid:1)
and a similar estimate holds true in the case N =2.
Step 3: u (·−x ) converges Wechoosex suchthatliminf |u (·−x )|p >0,itsweaklimit(which
n n n n→+∞ˆB1 n n
exists because Step 1 ensures boundedness) must be some u6≡0.
By Sobolev embeddings, one can show that (I ∗ F(u ))F′(u ) ⇀
α n n
n→+∞
(I ∗F(u))F′(u) in Lp RN . This easily yields that u solves (1.1)
α loc
(cid:0) (cid:1)
5
Proof of Theorem 2.1.
By Lemma 2.2, I admits a Pohoˇzaev-Palais-Smale sequence (un)n∈N at the
energylevelb. WeapplyLemma2.3tothelattersequence: ifthefirstalternative
occurred,then we wouldhave I(u ) → I(0)=0, contradicting Lemma 2.3.
n
n→+∞
Therefore, the second alternative must occur and in particular u 6≡ 0 solves
(1.1).
We conclude this section by showing that Theorem 2.1 is actually sharp in the
|u|p
caseofapowernonlinearityF(u)= ;inotherwords,wegiveanon-existence
p
result for all the values p not matching the assumptions of Theorem 2.1. To
show non-existence,we use a Pohoˇzaevidentity, which is a classicalpropertyof
solutions of (1.1).
Proposition 2.4.
Any solution u of (1.1) satisfied the Pohoˇzaev identity (1.6).
Theorem 2.5.
|u|p
If F(u) = then problem (1.1) admits a non-trivial solution if and only if
p
α N +α α
p ∈ 1+ , , with the latter condition to be read as p > 1 + if
N N −2 2
(cid:18) (cid:19)
N =2.
Proof.
α N +α |u|p
If p ∈ 1+ , then one can easily see that F(u) = , hence the
N N −2 p
(cid:18) (cid:19)
existence of non-trivial solutions follows from Theorem 2.1.
Conversely, assume p is outside that range and u solves (1.1), By testing both
sides of against u we get
|∇u|2+|u|2 = (I ∗|u|p)|u|p.
ˆ ˆ α
RN RN
(cid:0) (cid:1)
Moreover, u satisfies the Pohoˇzaevidentity (1.6), which has the form
N −2 N N +α
|∇u|2+ |u|2− (I ∗|u|p)|u|p =0.
2 ˆRN 2 ˆRN 2p ˆRN α
A linear combination of the two formulas gives
N −2 N +α N N +α
− |∇u|2+ − |u|2,
(cid:18) 2 2p (cid:19)ˆRN (cid:18) 2 2p (cid:19)
α N +α
which implies u≡0 if p≤1+ or p≥ .
N N −2
6
3 From solutions to groundstates
In the last part of this paper we show that the mountain pass solutions given
by Theorem 2.1 are actually energy-minimizing, in the sense of (1.7).
Theorem 3.1.
The mountain-pass solution found in Theorem 2.1 is actually a ground state,
namely its energy level is given by (1.7).
The previous Theorem can be easily proved by constructing, for any solution v
of (1.1), a path γ ∈Γ which attains its maximum energy on v.
v
Lemma 3.2.
Assume F satisfies (F ) and v ∈H1 RN \{0} solves (1.1). Then, there exists
1
a path γ ∈Γ such that:
v
(cid:0) (cid:1)
1 1
γ(0)=0 γ =v I(γ (t))<I(v) for any t6= I(γ (t))<0
v v v
2 2
(cid:18) (cid:19)
Proof.
Fix a non-trivial solution v of (1.1) and consider the path γ =: [0,+∞) →
v
·
v if t>0
H1 RN defined by γ (t)= t .
v
( 0(cid:16) (cid:17) if t=0
Alon(cid:0)g th(cid:1)e path, the energy is given by (2.2), which is negative for t ≫ 1.
Moreover,due to the Pohoˇzaevidentity (1.6) we can also write
tN−2 N −2 tN N
I(γ (t))= − tN+α |∇u|2+ − tN+α |u|2,
v (cid:18) 2 2(N +α) (cid:19)ˆRN (cid:18) 2 2(N +α) (cid:19)ˆRN
whichhasitsmaximumint=1. Therefore,uptoarescalingoft,this pathhas
all the properties required.
Anyway, being
|∇γ (t)|2+|γ (t)|2 =tN−2 |∇v|2+tN |v|2,
ˆ v v ˆ ˆ
RN RN RN
(cid:0) (cid:1)
γ is continuous at t = 0 only if N ≥ 3, so in the case N = 2 we need a
v
modification for t close to 0.
·
v if t>t
t 0
If N = 2 we take γv(t) =  t(cid:16) (cid:17) · for some suitable t0 ≪ 1.
v if t≤t
 t t 0
0 (cid:18) 0(cid:19)
We only need to verify that I(γ (t))≤I(γ (1)) for t≤t .
Using the assumption(F′) andMv oser-Trudvinger’s(2.1)a0nd Hardy-Littlewood-
1
Sobolev inequalities we get
(I ∗F(γ (t)))F(γ (t))≤C R2|γv(t)|2 1+α2 =Ct2+α |v|2 1+α2 ,
ˆR2 α v v (cid:18) ´R2|∇γv(t)|2(cid:19) 0 (cid:18)ˆR2 (cid:19)
´
therefore using again Pohoˇzaevidentity we get, for t small enough,
0
1 t2 1+α2
I(γ (t)) ≤ |∇v|2+ 0 |v|2+Ct2+α |v|2
v 2ˆR2 2 ˆR2 0 (cid:18)ˆR2 (cid:19)
7
t2 α 1+α2
= I(v)+ 0 − |v|2+Ct2+α |v|2 <I(v)
(cid:18)2 2(2+α)(cid:19)ˆR2 0 (cid:18)ˆR2 (cid:19)
and the proof is complete.
Proof of Theorem 3.1.
Let u be the mountain-pass solution found in Theorem 2.1. By the lower-
semicontinuity of the norm we find I(u) ≤ b, whereas the definition (1.7) of
ground state yields I(u)≥c.
Now, take another solution v ∈H1 RN \{0} and apply Lemma 3.2: we get
I(v)= sup I(γ (t(cid:0)))≥(cid:1)inf sup I(γ(t))=b.
v
t∈[0,1] γ∈Γt∈[0,1]
Being v arbitrary, we get c ≥ b, hence c ≤ I(u) ≤ b ≤ c, therefore I(u) = b =
c.
References
[1] AdachiS.,TanakaK.TrudingertypeinequalitiesinRN andtheirbestexponents,Proc.
Amer.Math.Soc.128(2000), no.7,2051-2057
[2] Battaglia L., Van Schaftingen J.Existence of groundstates for a class of nonlinear
Choquard equations inthe plane,Adv.NonlinearStud.,accepted
[3] Berestycki, H., Gallou¨et ,T., Kavian,O. E´quations de champs scalaires euclidiens
nonlin´eairesdansleplan,C.R.Acad.Sci.PariesS´er.IMath.297(1983),no.5,307-310
[4] Berestycki,H.,LionsP.-L.Nonlinearscalarfieldequations. II.Existenceofinfinitely
many solutions, Arch.RationalMech.Anal.82(1983), no.4,347-375
[5] Jeanjean, L. Existence of solutions with prescribed norm for semilinear elliptic equa-
tions,NonlinearAnal.28(1997), no.10,1633-1659
[6] Lions, P.-L., The Choquard equation and related questions, Nonlinear Anal. 4 (1980),
no.6,1063-1072
[7] Moroz V., Van Schaftingen J. Groundstates of nonlinear Choquard equations: exis-
tence, qualitative properties and decay asymptotics, J. Funct. Anal 265 (2013), no. 2,
153-184
[8] Moroz V., Van Schaftingen J. Existence of groundstates for a class of nonlinear
Choquard equations,Trans.Amer.Math.Soc.367(2015), no.9,6557-6579
[9] Willem, M.,Minimax theorems,ProgressinNonlinearDifferentialEquations andtheir
Applications,24,Birkh¨auser,Boston, Mass.1996
8