Schr\"odinger-Poisson systems with a general critical nonlinearity

0.21 MB·English

by  Jianjun Zhang

SCHRO¨DINGER-POISSON SYSTEMS WITH
A GENERAL CRITICAL NONLINEARITY
JIANJUNZHANG,JOA˜O MARCOS DO O´, ANDMARCO SQUASSINA
5
1 Abstract. WeconsideraSchr¨odinger-Poissonsysteminvolvingageneralnonlinearityatcritical
0
growth and we prove the existence of positive solutions. The Ambrosetti-Rabinowitz condition
2
is not required. We also studytheasymptotics of solutions with respect to a parameter.

c
e
D
3 1. Introduction and main result
1
We are concerned with the nonlinear Schro¨dinger-Poisson system

P] ∆u+u+λφu = f(u) in R3,
A (1.1) −∆φ= λu2, in R3,
(cid:26) −
.
h where λ > 0 and the nonlinearity f reaches the critical growth. In the last decade, the
t Schro¨dinger-Poisson system has been object of intensive research because of its strong relevance
a
m inapplications. Fromaphysicalpointofview,itdescribessystemsofidenticallychargedparticles
[ interacting each other in the case where magnetic effects can be neglected. The nonlinear term

f models the interaction between the particles and the coupled term φu concerns the interaction
2
with the electric field. For more detailed physical aspects of the Schro¨dinger-Poisson system,
v
0 we refer the reader to [1,7,8,30] and to the references therein. In recent years, there has been
1 an increasing attention towards systems like (1.1) and the existence of positive solutions, sign-
1
changing solutions, ground states, radial and non-radial solutions and semi-classical states has
1
0 been investigated. In [17], D’Aprile and Mugnai obtained the existence of a nontrivial radial
. solution to (1.1) with f(u) = up 2u, for p [4,6). In [18], D’Aprile proved that system (1.1)
1 −
| | ∈
0 admitsanon-radialsolutionforf(u) = up 2u,withp (4,6). In[3],byusingtheConcentration
−
5 CompactnessPrinciple,AzzolliniandP|om|ponioobtain∈edtheexistenceofagroundstatesolution
1
to (1.1) with f(u)= u p 2u, for p (3,6). In [26], Ruiz obtained some nonexistence results for
: −
v | | ∈
Xi (1.2) −∆∆φu+=uu2+, λφu = |u|p−2u iinnRR33,
r (cid:26) −
a and established the relation between the existence of the positive solutions to system (1.2) and
the parameters p (2,6) and λ > 0. Moreover, if λ 1, the author showed that p = 3 is a
∈ ≥ 4
critical value for the existence of the positive solutions. For p (2,3), Ruiz [28] investigated
∈
the existence of radial ground states to system (1.2) and obtained the different behavior of the
solutions depending on p as λ 0. We also would like to cite some works [19,27], where system
→
(1.2) was considered as λ 0. In [19,27], the authors were concerned with the semi-classical
→
states for system (1.2). Precisely, the authors studied the existence of radial positive solutions
concentrating around a sphere. Recently, some works were focused on the existence of sign-
changing solutions to (1.1) with f(u) = up 2u. By using a gluing method, Kim and Seok [24]
−
| |
2000 Mathematics Subject Classification. 35B25, 35B33, 35J61.
Key words and phrases. Schr¨odinger-Poisson systems, variational methods, critical growth.
Research partially supported by INCTmat/MCT/Brazil. J.M. do O´ was supported by CNPq, CAPES/Brazil,
Jianjun Zhangwas partially supported byCAPES/Brazil and CPSF (2013M530868).
1
2 J.ZHANG,J.M.DOO´,ANDM.SQUASSINA
proved the existence of sign-changing solutions with a prescribed number of nodal domains for
(1.1) with p (4,6). Subsequently, Ianni [20] obtained a similar result for p [4,6). More
∈ ∈
recently, Wang and Zhou [31] considered the non-autonomous system
∆u+V(x)u+λφu = u p 2u in R3,
(1.3) −∆φ= u2, | | − in R3.
(cid:26) −
Undersuitable conditions on V, they proved theexistence of least energy sign-changing solutions
to system (1.3) with p (4,6) by minimizing over the sign-changing Nehari manifold. For
∈
furtherworks onthenon-autonomous Schro¨dinger-Poisson system, wealso wouldlike to mention
[2,14,22,25,32] and the references therein.
The works discussed above mainly focus on the study of system (1.1) with the very special
nonlinearity f(u) = up 2u. In [4], Azzollini, d’Avenia and Pomponio were concerned with the
−
| |
existence of apositiveradialsolution to system (1.1)undertheeffect of ageneral nonlinearterm,
see also [5,21]. Precisely, let g(u) = u+f(u), then
−
Theorem A (see [4]). Suppose
(H1) g(s) C(R,R);
∈
g(s) g(s)
(H2) < liminf limsup = m < 0;
−∞ s 0 s ≤ s 0 s −
g(→s) →
(H3) limsup 0;
s5 ≤
s
→∞ ξ
(H4) there exists ξ > 0 such that G(ξ ):= g(s)ds > 0.
0 0 0
Then there exists λ > 0 such that (1.1) admits a positive radial solution for λ (0,λ ).
0 R ∈ 0
(H )-(H )areknownasBerestycki-Lionsconditions,introducedin[9]. There,theauthorsshowed
1 4
that these conditions are almost necessary and sufficient for the existence of ground states to the
nonlinear scalar field equation ∆u= g(u), with u H1(RN), N 3.
− ∈ ≥
We remark that in the literature described above, only the subcritical case was considered. A
natural question arises on whether results like Theorem A holds if f is at critical growth. In fact,
in [33], Zhang, obtained the following
Theorem B (see [33]). Suppose f C(R,R) is odd and
∈
f(s)
(g ) lim = 0,
1 s
s 0
→ f(s)
(g ) lim = K > 0,
2 s + s5
(g ) T→her∞e exists D > 0 and q (2,6) such that f(s) Ks5+Dsq 1, for all s > 0,
3 −
∈ s ≥
(g ) There exists γ > 2 such that 0< γ f(τ)dτ sf(s), for all s = 0,
4 0 ≤ 6
Then (i) (1.1) has a positive radial solution for small λ > 0 if q (2,4] with D large enough, or
R ∈
q (4,6); (ii) if γ > 3, (1.1) admits a ground state solution for any λ > 0 provided q (2,4]
∈ ∈
with D large enough, or q (4,6).
∈
The author was able to obtain this existence result via a truncation argument. Condition
(g ) implies that f is superlinear and is the so-called Ambrosetti-Rabinowitz condition, usually
4
involved in guaranteeing the boundedness of (PS)-sequences.
The aim of this paper is to study the existence of the positive solutions to system (1.1) involving
a more general critical nonlinearity compared to that allowed in Theorem B. In particular, the
Ambrosetti-Rabinowitz condition is not required.
We shall assume that the following hypotheses on f:
(f ) f :R R is continuous, f = 0 on R and lim f(s) = 0.
1 → − s 0 s
→
SCHRO¨DINGER-POISSON SYSTEMS AT CRITICAL GROWTH 3
f(s)
(f ) limsup 1.
2 s5 ≤
s +
(f ) T→her∞e exist µ > 0 and q (2,6) such that f(s) µsq 1 for all s 0.
3 −
∈ ≥ ≥
Assumption (f ) implies f has (possibly) a critical growth at infinity and the limit of f(s)/s5 at
2
+ may fail to exist. Moreover, there exists κ > 0 such that
∞
1
(1.4) f(s) s+κs5 for all s 0.
≤ 2 ≥
Before stating the main result, we fix some notations. In the sequel, and denote the best
q
constants of Sobolev embeddings 1,2(R3) ֒ L6(R3) and H1(R3)֒ LSq(R3),C
D → →
1
3
u6dx u2dx, for all u 1,2(R3),
S R3| | ≤ R3|∇ | ∈ D
(cid:18)Z (cid:19) Z
2
q
uqdx ( u2+ u2)dx, for all u H1(R3).
q
C R3| | ≤ R3 |∇ | | | ∈
(cid:18)Z (cid:19) Z
Our main result is the following
Theorem 1.1. Suppose that f satisfies (f )-(f ).
1 3
(i) There exists λ > 0 such that, for every λ (0,λ ), system (1.1) admits a nontrivial
0 0
∈
positive solution (u ,φ ), provided that
λ λ
q−2
2
3q 6 q
µ > − 2;
" 2q 32 # Cq
S
(ii) Along a subsequence, (u ,φ ) converges to (u,0) in H1(R3) 1,2(R3) as λ 0, where
λ λ
×D →
u is a ground state solution to the limit problem
∆u+u = f(u), u H1(R3).
− ∈
The rest of the paper is devoted to prove Theorem 1.1. Since we are concerned with system
(1.1) with a more general nonlinear term f, the problem becomes more thorny and tough in
applying variational methods. In fact, due to the lack of the Ambrosetti-Rabinowitz condition,
the boundedness of (PS)-sequence is not easy to be obtained. To overcome this difficulty, we
will adopt a local deformation argument from Byeon and Jeanjean [10] to get a bounded (PS)-
sequence. Due to the presence of the nonlocal term φu, a crucial modification on the min-max
value is needed. We will define another min-max value C (see Section 3), where all paths are
λ
required to be uniformly bounded with respect to λ. Similar arguments can be found in [15].
The paper is organized as follows.
In Section 2 we consider the functional framework and some preliminary results.
In Section 3 we construct the min-max level.
In Section 4, we use a local deformation argument to give the proof of Theorem 1.1.
Notations.
u := updx 1/p for p [1, ).
p R3
• k k | | ∈ ∞
u := u 2 + u 2 1/2 for u H1(R3).
• k k (cid:0)kRk2 k∇ (cid:1)k2 ∈
H1(R3) is the subspace of H1(R3) of radially symmetric functions.
• 1r,2(R3(cid:0)):= u L2∗(R(cid:1)3) : u L2(R3) .
• D { ∈ ∇ ∈ }
4 J.ZHANG,J.M.DOO´,ANDM.SQUASSINA
2. Preliminaries and functional setting
We recall that, for u H1(R3), the Lax-Milgram theorem implies that there exists a unique
φ 1,2(R3) such tha∈t ∆φ= λu2 with
u
∈ D −
u2(y)
(2.1) φ (x) := λ dy.
u
R3 4π x y
Z | − |
Setting
1
T(u) := φ u2dx,
u
4 R3
Z
we summarize some properties of φ ,T(u), which will be used later.
u
Lemma 2.1 (see [26]). For any u H1(R3), we have
∈
(1) φ :H1(R3) 1,2(R3) is continuous and maps bounded sets into bounded sets.
u
7→ D
(2) φ 0,T(u) cλ u 4 for some c > 0.
u
(3) If u≥ u wea≤kly iknkH1(R3), then φ φ weakly in 1,2(R3).
(4) If un → u weakly in H1(R3), then Tu(nu→) =uT(u)+T(uD u)+o(1).
n n n
→ −
(5) If u is a radial function, so is φ .
u
Substituting (2.1) into (1.1), we can rewrite (1.1) in the following equivalent equation
(2.2) ∆u+u+λφ u= f(u), u H1(R3).
u
− ∈
We define the energy functional Γ : H1(R3) R by
λ
→
1 λ
Γ (u) = ( u2+u2)dx+ φ u2dx F(u)dx,
λ u
2 R3 |∇ | 4 R3 − R3
Z Z Z
with F(t) = tf(s)ds. It is standard to show that Γ is of class C1 on H1(R3). Since we are
0 λ
concerned with the positive solutions of (1.1), from now on, we can assume that f(s) = 0 for
R
everys 0. ItisreadilyprovedthatanycriticalpointofΓ isnonnegativeand,bythemaximum
λ
principl≤e, it is strictly positive. Moreover, it is easy to verify that (u,φ) H1(R3) 1,2(R3) is
a solution of (1.1) if and only if u H1(R3) is a critical point of the fu∈nctional Γ×.DIf λ = 0,
λ
∈
problem (2.2) becomes
(2.3) ∆u+u= f(u), u H1(R3),
− ∈
which will be referred as the limit problem of (2.2). In general, if a problem is well-behaved and
undergoes a small perturbation, then one may expect that the perturbed problem has a solution
near the solutions of the original problem. Then if λ is small, it is natural to find a solution of
(2.2) in some neighborhood of the solutions to the limit problem (2.3), which will play a crucial
roˇle in the study of perturbed problem (2.2). In the following, we study some properties of the
limit problem (2.3). First, we show the existence of the ground states of the limit problem (2.3).
Proposition 2.2. Suppose that f satisfies (f )-(f ), then the limit problem (2.3) has a ground
1 3
state u H1(R3), provided that
∈ r
q−2
2
3q 6 q
(2.4) µ > − 2.
" 2q 32 # Cq
S
Remark 2.3. In [6] the authors established the existence of the ground state solutions for the
nonlinearscalarfieldequationinvolvingcriticalgrowthinRN,forN 2. Inparticular,assuming
≥
that f satisfies (f )-(f ) and an additional condition
1 3
s
(f ) sf(s) 2F(s) 0 for all s 0, where F(s)= f(τ)dτ,
4 − ≥ ≥ 0
R
SCHRO¨DINGER-POISSON SYSTEMS AT CRITICAL GROWTH 5
the authors proved that (2.3) has a ground state. We remark that (f ) can be removed.
4
To prove Proposition 2.2, we will use the following notations.
:= u H1(R3) 0 : G(u)dx = 1 ,
M ∈ r \{ } R3
(cid:26) Z (cid:27)
:= u H1(R3) 0 :6 G(u)dx = u2dx ,
P ∈ r \{ } R3 R3|∇ |
(cid:26) Z Z (cid:27)
where G(t) = F(t) 1t2. is the so-called Pohozaˇev manifold. It follows, from (f ), that there
− 2 P 3
exists ξ > 0 such that G(ξ) > 0. Then it is easy to check that = and = . Define
M6 ∅ P 6 ∅
1
M := inf u2dx, p := inf I(u),
2 u R3|∇ | u
∈MZ ∈P
and the Mountain Pass value
b := inf max I(γ(t)),
γ Υ0 t 1
∈ ≤ ≤
where Υ = γ C([0,1],H1(R3)) :γ(0) = 0, I(γ(1)) < 0 and
{ ∈ r }
1
I(u) := ( u2 + u2)dx F(u)dx.
2 R3 |∇ | | | − R3
Z Z
Lemma 2.4. Let f satisfy (f1)-(f3) and (2.4). Then 0 < M < √326S and p < 31S32.
Proof. Obviously M [0, ). We claim that M > 0. Assume by contradiction that it is M = 0.
∈ ∞
Then there exists u such that u 0 as n . By Sobolev’s embedding
n n n 2
{ } ⊂ M k∇ k → → ∞
theorem, u 0 as n . Thus, it follows from (1.4) that
n 6
k k → → ∞
κ
limsup G(u )dx limsup u 6dx = 0,
n n
n R3 ≤ n 6 R3| |
→∞ Z →∞ Z
a contradiction, proving the claim. We now claim that p b. It suffices to prove that
≤
γ([0,1]) = for all γ Υ,
∩P 6 ∅ ∈
whose proof is similar to that in [23, Lemma 4.1]. Let
P(u) = u2dx 6 G(u)dx.
R3|∇ | − R3
Z Z
Then by (1.4) it is easy to know that there exists ρ > 0 such that
0
(2.5) P(u) > 0 if 0 < u ρ .
0
k k ≤
For any γ Υ, P(γ(0)) = 0 and P(γ(1)) 6I(γ(1)) < 0. Thus, there exists t (0,1) such
0
∈ ≤ ∈
that P(γ(t )) = 0 with γ(t ) > ρ , which implies γ([0,1]) = . We now use an idea from
0 0 0
Coleman-Glazer-Martink[16] tok prove that p = 2√3M32. Defi∩nePΦ6 : ∅ : (Φ(u))(x) = u(x),
9 M → P tu
where t = √6/6 u . Then Φ is a bijection. For u H1(R3), let us set
u 2
k∇ k ∈
1
T (u) = u2dx, V(u) = G(u)dx.
0
2 R3|∇ | R3
Z Z
Then for u , I(Φ(u)) = t T (u) t3V(u) = √6/18 u 3. Thus,
∈ M u 0 − u k∇ k2
inf I(u) = inf I(Φ(u)) = √6/18 inf u 3,
u u u k∇ k2
∈P ∈M ∈M
6 J.ZHANG,J.M.DOO´,ANDM.SQUASSINA
which implies p = 2√3M32. Finally, similar to that in [6], taking ψ H1(R3) with ψ 0 with
9 ∈ r ≥
ψ 2 = 1 and ψ = 1, then
k kq Cq− k k
t2 tq q 2 2 q
b ≤ mta0xI(tψ) ≤ mta0x 2 −µq kψkqq = 2−q µ−q−2Cqq−2.
≥ ≥ (cid:16) (cid:17)
Thus, by virtue of (2.4), we have p < 1 32 and, in turn, M < √36 . (cid:3)
3S 2 S
In the following, we will show that p can be achieved. This implies that the limit problem (2.3)
admits a ground state solution. Similar to that in [9], it is enough to prove that M can be
achieved. Now, we give the following Brezis-Lieb Lemma.
Lemma 2.5. Let h C(R3 R) and suppose that
∈ ×
h(x,t) h(x,t)
(2.6) lim = 0 and limsup | | < ,
t 0 t t 5 ∞
→ |t|→∞ | |
uniformly in x R3. If u u weakly in H1(R3) and u u a.e. in R3, then
n 0 n 0
∈ → →
(H(x,u ) H(x,u u ) H(x,u ))dx = o(1),
n n 0 0
R3 − − −
Z
t
where H(x,t) = h(x,s)ds.
0
Proof. The prooRf is standard. For the subcritical case, we refer to V. Coti Zelati and P.H.
Rabinowitz [13]. For any fixed δ > 0, set Ω (δ) := x R3 : u (x) u (x) δ . Then
n n 0
{ ∈ | − | ≤ }
(H(x,u ) H(x,u u ) H(x,u ))dx
n n 0 0
R3 − − −
Z
= (H(x,u ) H(x,u u ) H(x,u ))dx
n n 0 0
ZR3 Ωn(δ) − − −
\
+ (H(x,u ) H(x,u ))dx H(x,u u )dx
n 0 n 0
− − −
ZΩn(δ) ZΩn(δ)
:=J +J +J .
1 2 3
By conditions (2.6), for any ρ > 0, there exists C > 0 such that h(x,t) ρt +C t 5 for all
ρ ρ
(x,t) R R3. Then | | ≤ | | | |
∈ ×
ρ C
J u u 2+ ρ u u 6 dx
3 n 0 n 0
| |≤ 2| − | 6 | − |
ZΩn(δ)(cid:18) (cid:19)
ρ C
+ ρδ4 u u 2dx,
n 0
≤ 2 6 R3| − |
(cid:18) (cid:19)Z
and
J ρ(u + u )+C (u + u )5 u u dx
2 n 0 ρ n 0 n 0
| | ≤ | | | | | | | | | − |
ZΩn(δ)
(cid:2) 1 (cid:3) 1
2 2
ρ (u + u )2dx u u 2dx
n 0 n 0
≤ R3 | | | | R3| − |
(cid:18)Z (cid:19) (cid:18)Z (cid:19)
1
5
6
6
+C (u + u )6dx u u 6dx
ρ n 0 n 0
(cid:18)ZR3 | | | | (cid:19) ZΩn(δ)| − | !
SCHRO¨DINGER-POISSON SYSTEMS AT CRITICAL GROWTH 7
1 1
2 2
ρ (u + u )2dx u u 2dx
n 0 n 0
≤ R3 | | | | R3| − |
(cid:18)Z (cid:19) (cid:18)Z (cid:19)
5 1
2 6 6
+Cρδ3 (un + u0 )6dx un u0 2dx .
R3 | | | | R3| − |
(cid:18)Z (cid:19) (cid:18)Z (cid:19)
Then, since u is bounded in H1(R3), for every ε > 0, there exist ρ,δ > 0 such that
n n
{ }
J + J ε/2, for all n 1. On the other hand,
2 3
| | | | ≤ ≥
J = (H(x,u ) H(x,u u ) H(x,u ))dx
1 n n 0 0
− − −
ZBR(0)\Ωn(δ)
+ (H(x,u ) H(x,u u ) H(x,u ))dx
n n 0 0
ZR3\(Ωn(δ)SBR(0)) − − −
:= K +K ,
1 2
where B (0) = x RN : x < R ,R > 0. Noting that
R
{ ∈ | | }
K ρ(u + u )+C (u + u )5 u dx+ H(x,u )dx,
2 n 0 ρ n 0 0 0
| |≤ ZR3\BR(0) | | | | | | | | | | ZR3\BR(0)
(cid:2) (cid:3)
there exists R > 0 with K ε/4, for all n 1. Recall that u u a.e. in R3, then it follows
2 n 0
| |≤ ≥ →
from the Severini- Egoroff theorem that u converges to u in measure in B (0), which imples
n 0 R
lim B (0) Ω (δ) = 0.
R n
n 0| \ |
→
In turn K ε/4 for n large. Then J ε/2 for n large and the proof is complete. (cid:3)
1 1
| | ≤ | |≤
Proof of Proposition 2.2. The proof is similar to that of [34]. We may assume that there
exists u H1(R3) such that G(u )dx = 1 and u 2dx 2M, as n . By (f )-
{ n}⊂ r R3 n R3|∇ n| → → ∞ 1
(f ), u is bounded in H1(R3). Thus there is u H1(R3) such that, up to a subsequence,
u 2 {un}weakly in H1(R3).rTheRn 0 ∈R r
n → 0 r
u 2dx = u 2dx+ u u 2dx+o(1).
n 0 n 0
R3|∇ | R3|∇ | R3|∇ −∇ |
Z Z Z
Moreover, by Lemma 2.5, we have
G(u )dx = G(u )dx+ G(u u )dx+o(1).
n 0 n 0
R3 R3 R3 −
Z Z Z
It is easy to know that M = inf T (u) :V(u) = 1,u H1(R3) . Moreover,
{ 0 ∈ r }
T (u ) = T (v )+T (u )+o(1), V(u ) = V(v )+V(u )+o(1),
0 n 0 n 0 0 n n 0
where v = u u . Set S = T (v ),S = T (u ),V(v ) = λ ,V(u ) = λ , we have
n n 0 n 0 n 0 0 0 n n 0 0
−
λ = 1 λ + o(1) and S = M S + o(1). To prove that u is a minimizer of M, it
n 0 n 0 0
suffices t−o prove λ = 1, which implie−s u u strongly in H1(R3). It is easy to see that
0 n → 0 r
(2.7) T (u) M(V(u))1/3,
0
≥
for all u H1(R3) and V(u) 0. As we can see in [34], λ [0,1]. If λ [0,1), then λ > 0
0 0 n
∈ ≥ ∈ ∈
for n large enough. By (2.7), we have that S M(λ )1/3 and S M(λ )1/3. This implies
0 0 n n
≥ ≥
M = lim (S +S ) lim M (λ )1/3 +(λ )1/3
0 n 0 n
n ≥ n
→∞ →∞
(cid:16) (cid:17)
= M (λ )1/3 +(1 λ )1/3 M(λ +1 λ )= M,
0 0 0 0
− ≥ −
(cid:16) (cid:17)
8 J.ZHANG,J.M.DOO´,ANDM.SQUASSINA
which implies that λ = 0. So we get that u = 0 and lim S = M. By (f )-(f ), for any
0 0 n n 1 2
ε> 0, there exists C > 0 such that F(s) 1s2+C s4+(1+→∞ε)s6/6 for s R. Then
ε ≤ 4 ε ∈
1+ε
1 = lim λ limsup v 6,
n n ≤ 6 n k nk6
→∞ →∞
since v 0, namely limsup v 2 √3 6. Thus
k nk4 → n→∞k nk6 ≥
1 √36
M = limsup v 2 S limsup v 2 ,
2 k∇ nk2 ≥ 2 k nk6 ≥ 2 S
n n
→∞ →∞
which contradicts Lemma 2.4. Therefore, we conclude that λ = 1. Therefore, u and
0 0
∈ M
u 2dx = 2M. Setting t = u /√6, it follows from Coleman-Glazer-Martin [16] that
R3 0 0 0 2
ω =|∇u ( |) is a ground state sko∇lutiokn to problem (2.3). (cid:3)
R 0 t·0 ∈ P
Define as the set of the radial ground states U of (2.3). Then ω . Moreover, thanks to
r r
S ∈ S
Lemma 2.5, similarly as that in [11,23,34], we have the following
Proposition 2.6.
(i) b = I(ω), namely the Mountain Pass value agrees with the least energy level.
(ii) is compact in H1(R3).
Sr r
Proof. (i) Obviously, by (f )-(f ) we know that b is well defined. As we can see in the proof
1 3
of Lemma 2.4 and Proposition 2.2, we get that p b and p = I(ω). To prove b is the least
≤
energy, it suffices to prove b I(ω). Noting that ω is a ground state solution to (2.3), similar
≤
to that in [23], there exists a path γ Γ satisfying γ(0) = 0,I(γ(1)) < 0,ω γ([0,1]) and
∈ ∈
max I(γ(t)) = I(ω). Thus, b I(ω).
t [0,1]
∈ ≤
(ii) We adopt some ideas in [11] to show the compactness of . Similar to that in [11], is
r r
bounded in H1(R3). For any u , without loss of generaliSty, we can assume that u S u
r { n} ⊂ Sr n → 0
weakly in H1(R3) and u u a.e. in R3. It follows from [12] that v () := u ( M/3 ) is
a minimizerrof T (v) onn v→ H0 1(R3) : V(v) = 1 . This means that nv· is anpositive ·and
0 { ∈ r } { n} p
radially symmetric minimizing sequence of M. As we can see in the proof of Proposition 2.2,
v v := u ( M/3 ) strongly in H1(R3). Thus, u u strongly in H1(R3) and u ,
i.ne.,→ 0is com0pact. · r n → 0 r 0 ∈ S(cid:3)r
r
S p
3. The minimax level
Let U be arbitrary but fixed. By the Pohozaˇev identity, for U (x) = U(x) we have
∈ Sr t t
t t3
I(U ) = U 2dx.
t
2 − 6 R3|∇ |
(cid:16) (cid:17)Z
Thus, there exists t > 1 such that I(U ) < 2 for t t . Set
0 t 0
− ≥
D max Γ (U ).
λ λ t
≡ t [0,t0]
∈
Then, by virtue of Lemma 2.1, we get that D b, as λ 0.
λ
→ →
Moreover, it is easy to verify the following lemma, which is crucial to define the uniformly
bounded set of the mountain pathes as previously mentioned.
Lemma 3.1. There exist λ > 0 and > 0, such that for any 0 < λ < λ there hold
1 0 1
C
Γ (U )< 2, U , t (0,t ], u , u .
λ t0 − k tk≤ C0 ∀ ∈ 0 k k ≤ C0 ∀ ∈ Sr
SCHRO¨DINGER-POISSON SYSTEMS AT CRITICAL GROWTH 9
Proof. Due to the Pohozaˇev identity, as we can see in [11], there exists C > 0 such that u C
k k ≤
for any u . For U fixed above and t (0,t ],
r r 0
∈ S ∈ S ∈
U 2 = t U 2+t3 U 2 (t+t3) U 2 C2(t +t3).
k tk k∇ k2 k k2 ≤ k k ≤ 0 0
The second and last part of the assertion hold if = 2t2C. For the first part, by Lemma 2.1
C0 0
Γ (U ) I(U )+4cλ2 U 4 I(U )+4cλ2 4.
λ t0 ≤ t0 k t0k ≤ t0 C0
It follows from I(U ) < 2 that there exist λ > 0 with Γ (U )< 2 for any 0 < λ < λ . The
proof is completed.t0 − 1 λ t0 − 1 (cid:3)
Now, for any λ (0,λ ), we define a min-max value C :
1 λ
∈
C = inf max Γ (γ(s)),
λ λ
γ Υλs [0,t0]
∈ ∈
where
Υ = γ C([0,t ],H1(R3)) : γ(0) = 0,γ(t )= U , γ(t) +1,t [0,t ] .
λ ∈ 0 r 0 t0 k k ≤ C0 ∈ 0
Obviously, U (cid:8) Υ . Moreover, C D for λ (0,λ ). (cid:9)
t λ λ λ 1
∈ ≤ ∈
Proposition 3.2. lim C = b.
λ
λ 0
→
Proof. It suffices to prove that
liminfC b.
λ
λ 0 ≥
→
Noting that φ 0, we see that for any γ Υ , γ˜() = γ(t ) Υ. It follows that C b,
u λ 0 λ
concluding the p≥roof. ∈ · · ∈ ≥ (cid:3)
4. Proof of Theorem 1.1
Now for α,d > 0, define
Γα := u H1(R3): Γ (u) α
λ { ∈ r λ ≤ }
and
d = u H1(R3): inf u v d .
S (cid:26) ∈ r v∈Srk − k ≤ (cid:27)
Obviously, d, i.e., d = φ for all d> 0. For some 0 < d< 1, we will find a solution u d
r
S ⊂ S S 6 ∈ S
of problem (2.2) for sufficiently small λ > 0. The following proposition is crucial to obtain a
suitable (PS)-sequence for Γ and plays a key role in our proof. Choose
λ
1
1 3 4
(4.1) 0< d < min 3κ 1 ,√3b ,
−
3 2S
( )
(cid:20) (cid:21)
where κ is given in (1.4).
Proposition 4.1. Let λ be such that lim λ = 0 and for all i, u d with
{ i}∞i=1 i→∞ i { λi} ⊂ S
′
lim Γ (u ) b and lim Γ (u ) = 0.
i λi λi ≤ i λi λi
→∞ →∞
Then for d small enough, there is u , up to a subsequence, such that u u in H1(R3).
0 ∈ Sr λi → 0 r
10 J.ZHANG,J.M.DOO´,ANDM.SQUASSINA
Proof. For convenience, we write λ for λ . Since u d, there exist U and v H1(R3)
i λ λ r λ
∈ S ∈ S ∈
such that u = U + v with v d. Since is compact, up to a subsequence, there
λ λ λ λ r
exist U and v H1(R3)k, sukch≤that U SU strongly in H1(R3), v v weakly in
0 r 0 λ 0 λ 0
H1(R3), ∈vS d and∈v v a.e. in R3. Le→t u = U + v , then u →d and u u
0 λ 0 0 0 0 0 λ 0
weakly inkHk1(R≤3). It follow→s from lim Γ′ (u ) = 0 that I (u ) = 0. No∈w,Swe show u→ 0.
i→∞ λ λ ′ 0 0 6≡
Otherwise, if u 0, then U = v d. By (4.1), U < √3b. On the other hand, by
0 0 0 0
≡ k k k k ≤ k∇ k
U and the Pohozaev’s identity, U = √3b, which is a contradiction. So u 0
0 r 0 0
∈ S k∇ k 6≡
and I(u ) b. Meanwhile, thanks to Lemma 2.5, Γ (u ) = I(u ) + I(u u ) + o(1),
0 λ λ 0 λ 0
≥ −
then we have I(u u ) o(1). Thus, by (1.4) and the Sobolev’ embedding theorem,
λ 0
− ≤
u u 2 2κ 3 u u 6 + o(1). If u u 0 as λ 0, up to a subsequence,
k λ − 0k ≤ 3 S− k λ − 0k k λ − 0k 6→ →
1
we can get that u u 3 3κ 1 4 for λ small. This is a contradiction. Thus, u u
k λ − 0k ≥ 2S − λ → 0
strongly in H1(R3). The proof is completed. (cid:3)
r (cid:2) (cid:3)
By Proposition 4.1, there exist
1
1 3 4
0 < d< min 1, 3κ 1 ,√3b
−
3 2S
( )
(cid:18) (cid:19)
and ω > 0,λ0 > 0 such that kΓ′λ(u)k ≥ ω for u ∈ ΓDλλ (Sd \Sd2) and λ ∈ (0,λ0). Then, we
have the following proposition.
T
Proposition 4.2. There exists α > 0 such that for small λ > 0,
d
Γ (γ(s)) C α implies that γ(s) 2,
λ λ
≥ − ∈ S
where γ(s) = U( ),s (0,t ].
s· ∈ 0
Proof. From a change of variables and the Pohozaˇev identity,
s s3 λ
Γ (γ(s)) = U 2dx+ φ γ(s)2dx.
λ γ(s)
2 − 6 R3|∇ | 4 R3 | |
(cid:18) (cid:19)Z Z
It follows from Lemma 2.1 that Γ (γ(s)) = s s3 U 2dx+O(λ2). Note that
λ 2 − 6 R3|∇ |
s s(cid:0)3 (cid:1)R
max U 2dx = b
s∈[0,t0](cid:18)2 − 6 (cid:19)ZR3|∇ |
and C b as λ 0, the conclusion follows. (cid:3)
λ
→ →
Thenext proposition assures the existence of a boundedPalais-Smale sequence for Γ . Choosing
λ
small α > 0 satisfies
0
α 1
(4.2) α min , dω2 .
0
≤ {2 9 }
Noting that lim C = lim D = b, without loss of generality, we can assume that
λ 0 λ λ 0 λ
→ →
D < C +α 2C for λ > 0 small enough.
λ λ 0 λ
≤
Proposition 4.3. For any λ > 0 small enough, there exists u ΓDλ d such that
{ n}n ⊂ λ ∩ S
Γ (u ) 0 as n .
′λ n → → ∞
Proof. Assume by contradiction, there exists a(λ) > 0 such that Γ (u) a(λ), u d ΓDλ
| ′λ | ≥ ∈ S ∩ λ
for some small λ > 0. Then there exists a pseudo-gradient vector field T in H1(R3) on a
λ r