Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 150, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
MULTIPLE SOLUTIONS FOR SCHRÖDINGER-MAXWELL
SYSTEMS WITH UNBOUNDED AND DECAYING RADIAL
POTENTIALS
FANGFANG LIAO, XIAOPING WANG, ZHIGANG LIU
Abstract. This article concerns the nonlinear Schrödinger-Maxwell system
−∆u + V (|x|)u + Q(|x|)φu = Q(|x|)f (u),
2
−∆φ = Q(|x|)u ,
in R3
3
in R
where V and Q are unbounded and decaying radial. Under suitable assumptions on nonlinearity f (u), we establish the existence of nontrivial solutions
and a sequence of high energy solutions in weighted Sobolev space via Mountain Pass Theorem and symmetric Mountain Pass Theorem.
1. Introduction
This article concerns the nonlinear Schrödinger-Maxwell system
−∆u + V (|x|)u + Q(|x|)φu = Q(|x|)f (u),
−∆φ = Q(|x|)u2 ,
in R3 .
in R3
(1.1)
Such a system, also known as the nonlinear Schrödinger-Maxwell system, arises
in an interesting physical context. Indeed, according to a classical model, the
interaction of a charge particle with an electromagnetic field can be described by
coupling the nonlinear Schrödinger and the Maxwell equations. For more details on
the physical aspects, we refer to [1]. In particular, if we are looking for electrostatictype solutions, we just have to solve (1.1).
For this problem in a bounded domain, there are some works. Let us recall some
recent results. Benci and Fortunato obtained the existence of infinitely many solutions of an eigenvalue problem in [1]. D’Aprile and Wei [2] studied concentration
phenomena for the system in the unit ball B1 of R3 with Dirichlet boundary conditions. Candela and Salvatore [3] considered the problem with a non-homogeneous
term and obtained infinitely many radially symmetric solutions.
Recently, the problem in the whole space R3 was considered in some works, see
for instance [4 − 15] and the references therein. We recall some of them as follows.
2000 Mathematics Subject Classification. 35J20, 35J60.
Key words and phrases. Schrödinger-Maxwell system; unbounded or decaying potential;
weighted Sobolev space; mountain pass theorem.
c 2014 Texas State University - San Marcos.
Submitted March 14, 2014. Published June 27, 2014.
1
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F. LIAO, X. WANG, Z. LIU
EJDE-2014/150
Ruiz [4] considered the system
−∆u + V (x)u + λφu = Q(x)f (u),
−∆φ = u2 ,
in R3
in R3
(1.2)
and obtained the existence and nonexistence of radial solutions for (1.2) with
V (x) = Q(x) = 1, f (u) = up (1 < p < 5). Later, Ambrosetti and Ruiz in [5]
obtained multiplicity results for (1.2) with V (x) = Q(x) = 1. For the critical
growth case, we refer to [6]. Zhao and Zhao established the existence of a positive
solution by the concentration compactness principle. Sun, Chen and Nieto [7] obtained the existence of ground state solutions when V (x) = 1, λ = K(x) and f (u)
is asymptotically linear at infinity. When the potential V is not a constant, Wang
and Zhou [8] also considered that the case f (x, u) is asymptotically linear and the
positive potential V is bounded and non-radial. Mercuri [9] considered the potential
a
V may vanish at infinity and bounded; i.e., 1+|x|
α ≤ V (x) ≤ A for some α ∈ (0, 2],
a, A > 0. By using the classical Mountain Pass Theorem, the author obtained the
N +2
existence of positive solutions with λ = 1 and f (u) = up (1 < p < N
−2 , N = 3, 4, 5).
Soon after, Sun, Chen and Yang [10] considered the asymptotically linear case under the assumptions in [9], the existence and nonexistence of solutions are obtained
depending on the parameters λ. When λ = 1 and Q = 1, Chen and Tang [11]
considered the potential V (x) satisfies some coercive condition, i.e.,
(V0) V ∈ C(R3 , R) satisfies inf x∈R3 V (x) > 0 and for each M > 0, meas{x ∈
R3 |V (x) ≤ M } < +∞,
and proved that (1.2) has infinitely many high energy solutions under the condition
that f (x, u) is superlinear at infinity in u by fountain theorem established in [12].
Soon after, Li, Su and Wei [13] improved their results. For V (x) and f (x, u)
are 1-periodic in each x. Zhao and Zhao [14] considered this case and obtained
the existence of infinitely many geometrically distinct solutions. For the result of
semiclassical solutions, we refer to [17].
In the present paper, we will consider more general radial potential, that is, the
potential V (x) may be unbounded, decaying and vanishing. We make the following
assumptions:
(V1) V (r) ∈ C((0, +∞)), V (r) ≥ 0 and there exist a0 and a1 such that
lim inf
r→0
V (r)
> 0,
ra0
lim inf
r→+∞
V (r)
> 0,
ra1
(Q0) Q(r) ∈ C((0, +∞)), Q(r) ≥ 0 and there exist b0 and b1 such that
lim sup
r→0
Q(r)
> 0,
r b0
lim sup
r→+∞
Q(r)
> 0.
r b1
Next we introduce notation. Let C0∞ (R3 ) denote the collection of smooth functions
with compact support and
∞
C0,r
(R3 ) = {u ∈ C0∞ (R3 ) : u is radial}.
∞
Let Dr1,2 (R3 ) be the completion of C0,r
(R3 ) under the norm
Z
1/2
kukDr1,2 =
|∇u|2 dx
.
R3
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SCHRÖDINGER-MAXWELL SYSTEMS
3
Define
Lp (R3 ; Q) = {u : R3 → R; u is measurable and
Z
Q(|x|)|u|p dx < ∞}.
R3
with norm
kukp =
Z
Q(|x|)|u|p dx
1/p
.
R3
Set
E = Hr1 (R3 ; V ) = Dr1,2 (R3 ) ∩ L2 (R3 ; V ),
which is a Hilbert space with the norm
Z
1/2
.
kukE =
|∇u|2 + V (|x|)u2 dx
R3
Corresponding to [15], if (V1) and (Q0) are satisfied, for N = 3, we define


b0 ≥ −2, a0 ≥ −2,
6 + 2b0 ,
8+4b0 −2a0
p(a0 , b0 ) =
, −2 ≥ a0 > −4, b0 ≥ a0 ,
4+a0


∞,
a0 ≤ −4, b0 > −4,

8+4b1 −2a1

 4+a1 , b1 ≥ a1 > −2,
p(a1 , b1 ) = 6 + 2b1 ,
b1 ≥ −2, a1 ≤ −2,


2,
b1 ≤ max{a1 , −2}.
On the other hand, recently, Su, Wang and Willem [15] studied the nonlinear
Schrödinger equation
−∆u + V (|x|)u = Q(|x|)f (u),
u(x) → 0,
in RN
as |x| → ∞.
(1.3)
and assumed the Ambrosetti-Rabinowitz condition holds; i.e., there exists µ > 2
such that
0 < µF (u) ≤ uf (u),
∀u ∈ R,
Ru
where F (u) = 0 f (s)ds. They proved the existence of ground states solutions
when V and Q satisfy the assumption (V1) and (Q0).
Motivated by the above facts, as in [15], the purpose of this paper is to extend the
existence results of problem (1.3) to Schrödinger-Maxwell system (1.1). Moreover,
we assume
6p−12
(Q1) Q ∈ L 5p−12 (R3 ) for all p > 12/5.
To reduce our statement, we first make the following assumption on f .
(F1) f ∈ C(R, R), and |f (u)| ≤ c(|u|p1 −1 + |u|p2 −1 )
for some p < p1 ≤ p2 < p (p, p will be defined later), where c is a positive constant.
Throughout this article we denote by ci , Ci various positive constants, |·|p denotes
the usual Lp (R3 )-norm, and k · kq denotes the Lq (R3 , Q)-norm.
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F. LIAO, X. WANG, Z. LIU
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2. Preliminaries
To prove our results, we use the following lemma from [15].
Lemma 2.1. Assume (V1) and (Q0) with p = p(a0 , b0 ) ≥ p = p(a1 , b1 ). Then
Hr1 (R3 ; V ) ,→ Lp (R3 ; Q),
for p ≤ p ≤ p when p < ∞ and for p ≤ p < p when p = ∞. Furthermore, if b1 ≥
max{a1 , −2}, the embedding is compact for p < p < p, and if b1 < max{a1 , −2},
the embedding is compact for 2 ≤ p < p.
Remark 2.2. In particular, we can take p = p(a0 , b0 ) ≥ p = max{p(a1 , b1 ), 12/5}
if we take suitable a0 , b0 . Clearly, p = max{p(a1 , b1 ), 12
5 } ≥ p(a1 , b1 ). Thus, Lemma
2.1 holds for p=max{p(a1 , b1 ), 12/5}.
It is well known that system (1.1) is the Euler-Lagrange equation of the functional
J : E × Dr1,2 (R3 ) → R defined by
Z
Z
Z
1
1
1
J(u, φ) = kuk2E −
|∇φ|2 dx +
Q(|x|)φu2 dx −
Q(|x|)F (u)dx.
2
4 R3
2 R3
R3
For any u ∈ E, consider the linear functional Tu : Dr1,2 (R3 ) → R defined as
Z
Tu (v) =
Q(|x|)u2 vdx.
R3
For p < p < p, by (Q1), the Hölder inequality and Lemma 2.1, we have
Z
Q(|x|)u2 vdx
R3
Z
p−2
=
Q(|x|) p Q(|x|)2/p u2 vdx
R3
≤
Z
Q(|x|)
p−2
6p
p · 5p−12
5p−12
2/p Z
Z
6p
dx
(Q(|x|)2/p u2 )p/2 dx
R3
≤ S −1 |Q|
R3
p−2
p
6p−12
5p−12
≤ c1 S −1 |Q|
Z
p−2
p
6p−12
5p−12
R3
1/6
v 6 dx
R3
2/p
Q(|x|)up
kvkDr1,2
kuk2E kvkDr1,2 .
where S is the best Sobolev embedding constant. Hence, the Lax-Milgram theorem
implies that for every u ∈ E, there exists a unique φu ∈ Dr1,2 (R3 ) such that
Z
Z
Q(|x|)u2 v =
∇φu · ∇v, for any v ∈ Dr1,2 (R3 ),
R3
R3
Using the integration by parts, we obtain
Z
Z
∇φu ∇vdx = −
v∆φu dx,
R3
R3
for any v ∈ Dr1,2 (R3 );
therefore, −∆φu = Q(|x|)u2 . We can write an integral expression for φu in the
form
Z
1
Q(|y|)u2 (y)
dy,
(2.1)
φu =
4π R3 |x − y|
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SCHRÖDINGER-MAXWELL SYSTEMS
5
for any u ∈ C0∞ (R3 ), by density it can be extended for any u ∈ E. Moreover, the
functions φu possess the following properties:
kφu kDr1,2 ≤ c2 kuk2p ≤ c3 kuk2E .
φu ≥ 0,
In fact, clearly, φu ≥ 0 by (2.1). Using integration by parts, −∆φu = Q(|x|)u2 , the
Hölder inequality and the Sobolev inequality, for any u ∈ E, we obtain
Z
Z
kφu k2D1,2 =
∇φu · ∇φu dx = −
∆φu · φu dx
r
R3
R3
Z
=
Q(|x|)φu u2 dx
R3
p−2
p
kuk2p kφu kDr1,2
≤ c1 S −1 |Q| 6p−12
5p−12
≤
c2 |u|2p kφu kDr1,2 .
It follows that
kφu kDr1,2 ≤ c2 kuk2p ≤ c3 kuk2E .
Moreover, there exists c4 > 0 such that
Z
Q(|x|)φu u2 dx ≤ c4 kuk4E .
(2.2)
R3
So, we can consider the functional I : E → R3 defined by I(u) = J(u, φu ). By (2.1)
the reduced functional takes the form
Z
Z
1
1
2
2
Q(|x|)φu u dx −
Q(|x|)F (u)dx.
(2.3)
I(u) = kukE +
2
4 R3
R3
It is clear that I is well defined. Moreover, Our hypotheses imply that I ∈ C 1 (E, R)
and a standard argument shows that (u, φ) ∈ E × Dr1,2 (R3 ) is a critical point of J
if and only if u is a critical point of I and φ = φu (see [22]).
Lemma 2.3. If assumptions (V1), (Q0), (Q1), (F1) hold, then I ∈ C 1 (E, R) and
Z
Z
0
hI (u), vi =
(∇u · ∇v + V (|x|)uv)dx +
Q(|x|)φu uvdx − hΨ0 (u), vi, (2.4)
R3
where Ψ(u) =
R
R3
R3
Q(|x|)F (u)dx.
Proof. First, we prove the existence of the Gateaux derivative of Ψ. From (F1), we
have
|f (u)| ≤ c(|u|p1 −1 + |u|p2 −1 ),
1
1
|F (u)| ≤ c( |u|p1 + |u|p2 ).
p1
p2
(2.5)
(2.6)
For any u, v ∈ E and 0 < |t| < 1, by the mean value and (2.5), there exists 0 < θ < 1
such that
|Q(|x|)F (u + tv) − Q(|x|)F (u)|
|t|
= |Q(|x|)f (u + θtv)v|
≤ cQ(|x|)(|u + θtv|p1 −1 + |u + θtv|p2 −1 )|v|
≤ c5 Q(|x|)[(|u|p1 −1 |v| + |v|p1 ) + (|u|p2 −1 |v| + |v|p2 )]
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F. LIAO, X. WANG, Z. LIU
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The Hölder inequality implies
g(x) := cQ(|x|)[(|u|p1 −1 |v| + |v|p1 ) + (|u|p2 −1 |v| + |v|p2 )] ∈ L1 (R3 ).
Consequently, by the Lebesgue’s dominated convergence theorem, one has
Z
0
hΨ (u), vi =
Q(|x|)f (u)vdx.
R3
Next,we show that Ψ0 (·) : E → E ∗ is continuous. Assume that un → u in E. By
Lemma 2.1, we know that un → u in Lp (R3 ; Q), for p ≤ p ≤ p when p < ∞ and for
p ≤ p < p when p = ∞.
On the space Lp1 (R3 ; Q) ∩ Lp2 (R3 ; Q), we define the norm
kukp1 ∧p2 = kukp1 + kukp2
Z
1/p1 Z
=
Q(|x|)|u|p1 dx
+
R3
p1
3
1/p2
Q(|x|)|u|p2 dx
R3
p2
3
On the space L (R ; Q) + L (R ; Q), we define the norm
kukp1 ∨p2 = inf kvkp1 + kwkp2 : v ∈ Lp1 (R3 ; Q), w ∈ Lp2 (R3 ; Q), u = v + w .
Since p < p1 ≤ p2 < p, one has un → u in Lp1 (R3 ; Q) ∩ Lp2 (R3 ; Q). Similar to [22,
Theorem A.4], we have
f (un ) → f (u)
0
0
in Lp1 (R3 ; Q) + Lp2 (R3 ; Q).
By the Hölder inequality, we have
|hΨ0 (un ) − Ψ0 (u), vi| ≤ kf (un ) − f (u)kp01 ∨p02 kvkp1 ∧p2
≤ c6 kf (un ) − f (u)kp01 ∨p02 kvkE ,
where p0i = pi /(pi − 1), i = 1, 2. Hence
kΨ0 (un ) − Ψ0 (u)k ≤ c6 kf (un ) − f (u)kp01 ∨p02 → 0
0
as n → ∞.
∗
This shows Ψ (·) : E → E is continuous. This completes the proof.
Lemma 2.4. Under the condition (F1), if {un } ⊂ E is a bounded sequence with
I 0 (un ) → 0, then {un } has a convergent subsequence.
Proof. Since {un } ⊂ E is bounded and the embedding E ,→ Ls (R3 ; Q) is compact
for each s ∈ (p, p), passing to a subsequence, we can assume that un * u in E, and
un → u in Ls (R3 ; Q), s ∈ (p, p).
Note that
hI 0 (un ) − I 0 (u), un − ui
Z
= kun − uk2E +
Q(|x|)φun u2n − Q(|x|)φun un u dx
R3
Z
Z
2
+
Q(|x|)φu u − Q(|x|)φu un u dx −
Q(|x|) (f (un ) − f (u)) (un − u)dx.
R3
R3
We have
kun − uk2E
0
0
Z
= hI (un ) − I (u), un − ui −
R3
Q(|x|)φun u2n − Q(|x|)φun un u dx
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Z
SCHRÖDINGER-MAXWELL SYSTEMS
Z
Q(|x|)φu u2 − Q(|x|)φu un u dx +
−
7
Q(|x|) (f (un ) − f (u)) (un − u)dx.
R3
R3
Since un * u in E and I 0 (un ) → 0, we have
hI 0 (un ) − I 0 (u), un − ui → 0
as n → ∞.
On one hand, by (Q1), for p < p < p, we have
Z
Z
2
Q(|x|)(φun un − φun un u)dx =
Q(|x|)φun un (un − u)dx
R3
R3
p−2
p
kun − ukp kun kp kφun k6
≤ |Q| 6p−12
5p−12
≤ c7 kun − ukp kun kp kφun kDr1,2 .
Hence,
Z
R3
Q(|x|) φun u2n − φun un u dx → 0,
as n → ∞.
Similarly,
Z
Q(|x|) φu u2 − φu un u dx → 0,
as n → ∞.
R3
On the other hand,
Z
|
Q(|x|) (f (un ) − f (u)) (un − u)dx|
R3
Z
≤
Q(|x|) (|f (un )| + |f (u)|) |un − u|dx
R3
Z
≤c
Q(|x|) |un |p1 −1 + |un |p2 −1 + |u|p1 −1 + |u|p2 −1 |un − u|dx
R3
Z
≤c
p1
Q(|x|)|un − u| dx
1/p1 Z
R3
+
Q(|x|)|un |p1 dx
p1p−1
1
R3
p1p−1 1
Q(|x|)|u|p1 dx
Z
R3
Z
+c
Q(|x|)|un − u|p2 dx
1/p2 Z
R3
+
Q(|x|)|un |p2 dx
p2p−1
2
R3
p2p−1 2
Q(|x|)|u| dx
.
Z
p2
R3
Since un → u in Ls (R3 ; Q), s ∈ (p, p), we have
Z
Q(|x|) (f (un ) − f (u)) (un − u)dx → 0
as n → ∞.
R3
So we have kun − ukE → 0. This completes the proof.
3. Main results
Theorem 3.1. Assume that conditions (V1), (Q0), (Q1) hold. If (F1) and the
following conidition hold
(F2) There exists µ and r > 0 such that max{p, 4} < µ ≤ p < ∞, and
µF (u) ≤ uf (u), ∀u ∈ R,
inf F (u) := β > 0.
|u|=r
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F. LIAO, X. WANG, Z. LIU
EJDE-2014/150
Then system (1.1) has a nontrivial solution. Furthermore, if f (u) is odd in u, then
system (1.1) has a sequence {(un , φn )} of solutions in E ×Dr1,2 (R3 ) with kun k → ∞
and I(un ) → +∞.
Proof. From (F1), we have
|F (u)| ≤ c(
1 p1
1
|u| + |u|p2 ).
p1
p2
Note that
Z
Z
1
1
kuk2E +
Q(|x|)φu u2 dx −
Q(|x|)F (u)dx
2
4 3
R3
Z R
1
≥ kuk2E −
Q(|x|)F (u)dx
2
R3
c
c
1
≥ kuk2E − kukpp11 − kukpp22
2
p1
p2
1
≥ kuk2E − c8 kukpE1 − c9 kukpE2 .
2
Since p1 , p2 > 2, we can take a small ρ such that
1
I|∂Bρ ≥ ρ2 − c8 ρp1 − c9 ρp2 := δ > 0,
2
where Bρ = {u ∈ E : kukE < ρ}.
For z ∈ R, set
h(t) := F (t−1 z)tµ , ∀t ∈ [1, +∞).
For |z| ≥ r and t ∈ [1, |z|/r], by (F2), one has
z
h0 (t) = f (t−1 z)(− 2 )tµ + F (t−1 z)µtµ−1
t
= tµ−1 µF (t−1 z) − t−1 zf (t−1 z) ≤ 0.
I(u) =
So, we have
β
|z|
) ≥ µ |z|µ .
r
r
Since µ > 4, there exists a constant max{p, 4} < α < p such that α < µ, and hence
F (z) = h(1) ≥ h(
F (u)
= +∞.
|u|→∞ |u|α
lim
(3.1)
For any finite dimensional space E1 ⊂ E, by the equivalence of norms in the
finite space, there exists a constant c(α) > 0, such that
kukα ≥ cα kukE ,
∀u ∈ E1
(3.2)
where α is the constant appearing in (3.1). For any σ > 0, by (F1), there is a
constant cσ > 0 such that
|F (u)| ≤ cσ |u|p ,
∀|u| < σ.
Hence, by (3.1), we know that for M > 0, there is a constant CM > 0 such that
F (u) ≥ M |u|α − CM |u|p , ∀u ∈ R.
By (3.2) and (3.3), we have
1
c4
p
I(u) ≤ kuk2E + kuk4E − M kukα
α + CM kukp
2
4
(3.3)
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SCHRÖDINGER-MAXWELL SYSTEMS
9
1
c4
p
α
kuk2E + kuk4E − M cα
α kukE + CM kukE ,
2
4
for all u ∈ E1 . Consequently, there is a large r1 > 0 such that I < 0 on E1 \Br1 .
Consequently, there is a point e ∈ E with kekE > ρ such that I(e) < 0.
Now, we prove that I satisfies the Palais-Smale condition. By Lemma 2.4 we
know that it is sufficient to prove {un } is bounded in E. Indeed, if a sequence
{un } ⊂ E such that I(un ) is bounded and I 0 (un ) → 0, then there is positive
constant M0 such that for large n, one has
≤
1 0
hI (un ), un i
µ
Z
1
1
1
1
≥ ( − )kun k2E + ( − )
Q(|x|)φun u2n dx
2 µ
4 µ R3
Z
f (u )u
n n
− F (un ) dx
Q(|x|)
+
µ
R3
1
1
≥ ( − )kun k2E .
2 µ
M0 + kun kE ≥ I(un ) −
This implies {un } is bounded.
Obviously, I(0) = 0. Hence I possesses a critical value η ≥ δ by [20, Theorem
2.2], thus problem (1.1) has a nontrivial solution. Moreover, obviously, I is bounded
on each bounded subset of E and f (u) is odd which implies I is even. Hence the
second conclusion follows from [20, Theorem 9.12]. This completes the proof. Note that µ > 4 in condition (F2). Now, we consider the weak case µ = 4. At
this one, we have the following Theorem.
Lemma 3.2. Assume that conditions (V1), (Q0), (Q1), (F1) and the following
conditions hold:
(u)
(F3) F|u|
4 → +∞ as |u| → +∞.
(F4) uf (u) ≥ 4F (u) for all u ∈ R.
If p < 4 < p, then system (1.1) has at least one nontrivial solution. Furthermore,
if f (u) is odd in u, then system (1.1) has a sequence {(un , φn )} of solutions in
E × Dr1,2 (R3 ) with kun k → ∞ and I(un ) → +∞.
Proof. From the proofs of the first segment in Theorem 3.1, we know that there
exist constants ρ > 0 and δ > 0 such that
I|∂Bρ ≥ δ > 0.
Moreover, for any finite dimensional space E1 ⊂ E, by the equivalence of norms in
the finite space, there exists a constant C > 0, such that
kuk4 ≥ CkukE ,
∀u ∈ E1 .
(3.4)
c4
Since p < 4, by (F1) and (F3) we know that for any M > 4C
4 , there is a constant
CM > 0 such that
F (u) ≥ M |u|4 − c(M )|u|p , ∀u ∈ R.
(3.5)
Hence
1
1
I(u) ≤ kuk2E +
2
4
Z
R3
p
Q(|x|)φun u2n dx − M kuk44 + CM kukp .
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F. LIAO, X. WANG, Z. LIU
EJDE-2014/150
By (3.4) and (3.5), we know
1
c4
p
kuk2E + kuk4E − M C 4 kuk4E + CM kukE ,
2
4
for all u ∈ E1 . Consequently, there is a large r1 > 0 such that I < 0 on E1 \Br1 .
Consequently, there is a point e ∈ E with kekE > ρ, such that I(e) < 0.
Next we prove that I satisfies the Palais-Smale condition. Indeed, if a sequence
{un } ⊂ E is such that {I(un )} is bounded and I 0 (un ) → 0, then there is a positive
constant M1 such that for large n, one has
1
M1 + kun kE ≥ I(un ) − hI 0 (un ), un i
4 Z
1
1
2
= kun kE +
Q(|x|) f (un )un − F (un ) dx
4
4
R3
1
≥ kun k2E .
4
This implies {un } is bounded. Hence {un } ⊂ E has a convergent subsequence by
Lemma 2.4. This shows that I satisfies the Palais-Smale condition. Finally, the
conclusions follows from [20, Theorem 2.2 and 9.12].
I(u) ≤
Corollary 3.3. Assume that conditions (V1), (Q0), (Q1), (F1), (F3) and the following conditions hold:
(F4’) u → f (u)/|u|3 is increasing on (−∞, 0) and on (0, +∞).
If p < 4 < p, then system (1.1) has at least one nontrivial solution. Furthermore,
if f (u) is odd in u, then system (1.1) has a sequence {(un , φn )} of solutions in
E × Dr1,2 (R3 ) with kun k → ∞ and I(un ) → +∞.
Proof. It is sufficient to prove that (F4’) implies (F4). In fact, whenever u > 0,
Z 1
Z 1
Z 1
f (u) 4 3
1
f (ut) 4 3
u t dt ≤
u t dt = f (u)u.
F (u) =
f (ut)u dt =
3
3
(ut)
(u)
4
0
0
0
Whenever u < 0,
Z 1
Z
F (u) =
f (ut)u dt = −
0
This shows (F4) holds.
0
1
f (ut) 4 3
u t dt ≤
(−ut)3
Z
0
1
f (u) 4 3
1
u t dt = f (u)u.
(u)3
4
Theorem 3.4. Assume that condition (V1), (Q1), (F1), (F3) and the following
condition hold:
(F5) F (u) ≥ 0 for all u ∈ R and G(s) ≤ G(t) whenever (s, t) ∈ R+ × R+ and
s ≤ t, where G(u) = f (u)u − 4F (u).
If p < 4 < p, then system (1.1) has at least one nontrivial solution. Furthermore,
if f (u) is odd in u, then system (1.1) has a sequence {(un , φn )} of solutions in
E × Dr1,2 (R3 ) with kun k → ∞ and I(un ) → +∞.
Proof. Similar to the proof of Lemma 3.2, we know that there exist ρ > 0, δ > 0
such that
I|∂Bρ ≥ δ > 0.
Moreover, for any finite dimensional subspace E1 ⊂ E, there is a large r1 > 0 such
that I < 0 on E1 \Br1 .
EJDE-2014/150
SCHRÖDINGER-MAXWELL SYSTEMS
11
Now, we prove that I satisfies the Cerami condition. Indeed, if a sequence
{un } ⊂ E is such that {I(un )} is bounded and (1 + kun k)I 0 (un ) → 0, then we
claim that {un } is bounded. If this is false, then we can assume kun k → +∞. Set
vn = kuunnkE , then kvn kE = 1. By virtue of Lemma 2.1, passing to a subsequence,
we may assume
vn * v
s
in E,
3
un → u in L (R ; Q), s ∈ (p, p).
Since {I(un )} is bounded, there exists a constant C1 > 0 such that
Z
Q(|x|)F (un )
dx ≤ C1 < ∞.
kun k4E
R3
Set Ω = {x ∈ R3 : v(x) 6= 0}. Then |un (x)| → +∞ for a.e. x ∈ Ω. If meas(Ω) > 0,
then, by (F4)
F (un )
F (un )
=
|vn (x)|4 → ∞, as n → ∞.
kun k4E
|un |4
Since Q(|x|) > 0, using Fatou’s lemma, we obtain
Z
Q(|x|)F (un )
dx → ∞.
kun k4E
R3
A contradiction, so meas(Ω) = 0. Therefore, v(x) = 0 a.e. x ∈ R3 . Next, as in [19],
we define
I(tn un ) = max I(tun ).
t∈[0,1]
√
√
un
For any M > 0, set ṽn = 4M kun kE = 4M vn . Since |F (u)| ≤ c( p11 |u|p1 + p12 |u|p2 )
for u ∈ R,
Z
Z
Z
c
c
p1
|
Q(|x|)F (ṽn )dx| ≤
Q(|x|)|ṽn | dx +
Q(|x|)|ṽn |p2 dx → 0,
p1 R3
p2 R3
R3
as n → ∞. Consequently, for large n, one has
I(tn un ) ≥ I(ṽn )
Z
Z
1
1
Q(|x|)φṽn ṽn2 dx −
≥ kṽn k2E +
Q(|x|)F (ṽn )dx
2
4 R3
R3
≥ M.
This means that limn→∞ I(tn un ) = ∞. In view of the choice of tn we know that
hI 0 (tn un ), tn un i = 0 or → 0. Hence, by (F5) and the oddness of f , one has
∞ ← 4I(tn un ) − hI 0 (tn un ), tn un i
Z
Z
2
2
2
= tn
|∇un | + V (|x|)|un | dx +
Q(|x|) (f (tn un )tn un − 4F (tn un )) dx
R3
R3
Z
≤ kun k2E +
Q(|x|) (f (un )un − 4F (un )) dx
R3
= 4I(un ) − hI 0 (un ), un i.
This is a contradiction, so {un } is bounded. Consequently, {un } ⊂ E has a convergent subsequence by Lemma 2.4. This shows that I satisfies the Cerami condition.
Note that if we use Cerami condition in place of the Palais-Smale condition, then
12
F. LIAO, X. WANG, Z. LIU
EJDE-2014/150
[20, Theorems 2.2 and 9.12] are still true. Therefore, the conclusion follows from
[20, Theorems 2.2 and 9.12]. This completes the proof.
Acknowledgments. This work is partially supported by the NNSF (11171351) of
China, the National Natural Science Foundation of Hunan Province (14JJ2133),the
Scientific Research Foundation of Hunan Provincial Education Department (13A093),
Scientific Research Fund of Hunan Provincial Education Department (12C0895),
and the Construct Program of the Key Discipline in Hunan Province.
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EJDE-2014/150
SCHRÖDINGER-MAXWELL SYSTEMS
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Fangfang Liao
School of Mathematics and Statistics, Central South University, Changsha, 410083,
Hunan, China.
Department of Mathematics, Xiangnan University, Chenzhou, 423000, Hunan, China
E-mail address: liaofangfang1981@126.com
Xiaoping Wang
Department of Mathematics, Xiangnan University, Chenzhou, 423000, Hunan, China
E-mail address: wxp31415@163.com
Zhigang Liu
Department of Mathematics, Xiangnan University, Chenzhou, 423000, Hunan, China
E-mail address: liuzg22@sina.com