Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 91, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS TO QUASILINEAR
SCHRÖDINGER EQUATIONS WITH INDEFINITE POTENTIAL
ZUPEI SHEN, ZHIQING HAN
Abstract. In this article, we study the existence and multiplicity of solutions
of the quasilinear Schrödinger equation
−u00 + V (x)u − (|u|2 )00 u = f (u)
on R, where the potential V allows sign changing and the nonlinearity satisfies
conditions weaker than the classical Ambrosetti-Rabinowitz condition. By a
local linking theorem and the fountain theorem, we obtain the existence and
multiplicity of solutions for the equation.
1. Introduction
We study the existence and multiplicity of solutions for the quasilinear elliptic
equation
− u00 + V (x)u − (|u|2 )00 u = f (u), x ∈ R.
(1.1)
Solutions of the equation are related to standing wave solutions for quasilinear
Schrödinger equation of the form
i∂t z = −z 00 + Ṽ (x)z − (|z|2 )00 z − f˜(|z|2 )z
(1.2)
which arises in various fields of physics, like the theory of superfluids or in dissipative quantum mechanics, plasma physics, fluid mechanics and in the theory of
Heisenberg ferromagnets, etc. For further physical motivations and a more complete list of references, we refer to [6, 9, 11] and the references therein.
As far as we know, the first existence result for equation (1.1) by variational
methods is due to [11], where by a constrained minimization argument the authors
proved the existence of a positive ground state solution with an unknown Lagrange
multiplier λ in front of the nonlinear term. Ambrosetti and Wang [12] considered
the existence of positive solutions of perturbation to the equation with a particular
nonlinearity g(u) = up . Alves et al [1], considered the existence and concentration
of positive solutions as → 0 for a related equation with 2 . Some related problems
on R are also considered in [2].
There is also much work devoting to the corresponding high dimensional equation; e.g. see [6, 10, 8, 9]. The solutions of equation (1.1) correspond to the critical
2000 Mathematics Subject Classification. 37J45, 58E05, 34C37, 70H05.
Key words and phrases. Quasilinear Schrödinger equation; local linking; fountain theorem;
indefinite potential.
c
2015
Texas State University - San Marcos.
Submitted October 24, 2014. Published April 10, 2015.
1
2
Z. SHEN, Z. HAN
EJDE-2015/91
points of the functional on H 1 (R):
Z
Z
Z
1
2
2
Φ(u) =
u0 + V (x)u2 dx + u0 u2 dx − F (u)dx,
(1.3)
2 R
R
R
Rt
where F (t) = 0 f (s)ds. All the above mentioned papers require that the potential
V (x) is positive. So that the energy functional Φ possesses the mountain pass geometry. Therefore, the mountain pass lemma can be applied. In [13], the authors
consider the case which the potential V (x) allows sign-change. However, in order
to satisfy the conditions of mountain pass theory, they need additional conditions
on nonlinearity. The similar assumptions are added in [14] (See Remark ). The
aim of the paper is to investigate equation (1.1) where the potential V (x) can be
sign changing and theR nonlinearity does not need to satisfy Ambrosetti-Rabinowitz
2
condition. The term R u0 u2 dx in (1.3) is homogeneous of order 4 and non-convex,
it prevents the linking geometric structure of the energy functional under our assumption. Inspired by the recent work of Chen and Liu [5] we make use of the
local linking theory to overcome this difficulty. To state our main results, we list
the assumptions on f and V as follows.
(V1) The potential V (x) ∈ C(R) is bounded from below and µ(V −1 (−∞, M )) <
∞ for every M > 0.
(F1) There exists C > 0 and p > 2 such that
|f (t)| ≤ C|t|p−1
for all t ∈ R.
(F2) 4F (t) ≤ f (t)t and
F (t)
= +∞ for all t ∈ R.
t4
We are now ready to state our results.
lim
t→∞
Theorem 1.1. Suppose that (V1), (F1)–(F2) are satisfied and f is odd. Then
equation (1.1) has a sequence of solutions such that Φ(uk ) → +∞.
Theorem 1.2. Suppose that (V1), (F1)–(F2) are satisfied. Then equation (1.1)
has at least one nontrivial solution.
Remark 1.3. In [13, 14], the authors need assumptions (F2) and
(G2) Fe(x, u) := 41 f (x, u)u − F (x, u) ≥ 0, and there exist c0 > 0 and σ >
max{1, N2N
+2 } such that
|F (x, u)|σ ≤ c0 |u|2σ Fe(x, u)
for all (x, u) ∈ RN .
In this paper, (G2 ) is not needed.
Throughout this paper, the letters C and Ci denote positive constants, which
may be different from place to place. The usual norm in Lp (R) with 1 ≤ p ≤ +∞
is denoted by | · |p
2. Preliminaries
To overcome the non-compactness of the embedding H 1 (R) ,→ L2 (R), we consider a linear subspace X of H 1 (R):
Z
1
X := {u ∈ H (R) :
V̄ (x)u2 dx < ∞}
R
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QUASILINEAR SCHRÖDINGER EQUATIONS
3
equipped with the inner product
Z
hu, vi =
u0 v 0 + V̄ uvdx
R
and the corresponding norm kuk = hu, vi1/2 , where V̄ = V (x) + m > 1 for a fixed
positive number m. It is well known that the imbedding X ,→ L2 (R) is compact
under the condition (V1). See [4]. Therefore, the eigenvalues of the operator
S := −∆ + V
can be numbered as
−∞ < λ1 ≤ λ2 ≤ λ3 . . . , λl → ∞
and the corresponding eigenfunctions are denoted by φ1 , φ2 . . . . We assume that
0 ∈ (λl , λl+1 ) for some l > 1. Let
X − = span{φ1 , . . . , φl },
X + = (X − )⊥ .
Then X − and X + are the negative and positive spaces of the quadratic form
Z
1
2
u0 (x) + V (x)u2 (x)dx.
Q(u) =
2 R
. It is well known that there is a positive constant α > 0 such that
± Q(u) ≥ αkuk2 ,
u ∈ X ±.
(2.1)
Let
1
J(u) =
2
Z
02
2
Z
u + V (x)u dx −
R
Z
F (u)dx,
I(u) =
R
2
u0 u2 dx.
R
Then
Φ(u) = J(u) + I(u).
By the continuous imbedding H 1 (R) ,→ L∞ (R) and X ,→ H 1 (R), I is well defined
on X and
|I(u)| ≤ |u|2∞ kuk2H 1 ≤ Ckuk4 for all u ∈ X.
(2.2)
To verify that the functional Φ is C 1 , it is sufficient to prove this for I(u).
Lemma 2.1. I(u) belongs to C 1 in X.
The proof of the above lemma is similar to that of [11, Lemma 1]. We omit
it here. From the above discussions, the functional Φ(u) is a C 1 functional with
derivative given by
Z
Z
Z
0
0 0
02
2 0 0
(Φ (u), v) =
u v + V̄ uvdx +
2u uv + 2u u v dx − g(u)vdx
R
R
R
Z
Z
(2.3)
02
2 0 0
= hu, vi + 2u uv + 2u u v dx −
g(u)vdx,
R
R
where g(t) = f (t) + mt. To prove our main results, we need to introduce some
definitions and theorems.
Definition. We say that Φ ∈ C 1 (X) satisfies condition (PS) if any sequence
(un )⊂ X such that
Φ(un ) → c, Φ0 (un ) → 0
has a convergent subsequence.
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Z. SHEN, Z. HAN
EJDE-2015/91
For the proof of Theorem 1.1, we use the following fountain theorem by Bartsch
[3]. For k = 1, 2 . . . , let
Yk = span{φ1 , . . . φk },
Zk = span{φk , φ1+k . . . }.
(2.4)
1
Theorem 2.2. Assume that the even functional Φ ∈ C (X) satisfies the (PS)
condition, if there exists k0 > 0 such that for k ≥ k0 there exists ρk > rk > 0 such
that
(i) bk = inf u∈Zk ,kuk=rk Φ(u) → +∞ as k → ∞,
(ii) αk = maxu∈Yk ,kuk=ρk Φ(u) ≤ 0.
Then Φ has a sequence of critical points {uk } such that Φ(uk ) → +∞.
For the proof of Theorem 1.2, we will use the local linking theorem. Recall that
the definition of local linking at 0 with respect to the direct sum decomposition
X = X + ⊕ X − , if there is ρ > 0 such that for u ∈ X −
Φ(u) ≤ 0,
for u ∈ X − , kuk ≤ ρ
Φ(u) ≥ 0,
for u ∈ X + , kuk ≤ ρ.
(2.5)
Next, we consider two sequences of finite dimensional subspaces
X0± ⊂ X1± ⊂ · · · ⊂ X ±
such that
X ± = ∪n∈N Xn± .
For multi-index α = (α− , α+ ) ∈ N2 , we set Xα = Xα− ⊕ Xα+ and denote by Φα
the restriction of Φ on Xα . A sequence {αn } ⊂ N2 is admissible if, for any α ∈ N2 ,
there is m ∈ N such that α ≤ αn for n ≥ m, where for α, β ∈ N2 , α ≤ β means
α± ≤ β ± . Obviously, if {αn } is admissible, then any subsequence of {αn } is also
admissible.
Definition. We say that Φ ∈ C 1 (X) satisfies condition (C)∗ if, whenever {αn } ⊂
N2 admissible, any sequence {un } ⊂ X such that
un ∈ Xαn ,
sup Φ(un ) < ∞,
n
(1 + kun k)kΦ0αn (un )k → 0
(2.6)
contains a subsequence which converges to a critical point of Φ.
Theorem 2.3 (Local linking theorem [7]). Suppose that Φ ∈ C 1 (X) has a local
linking at 0, Φ satisfies (C)∗ , Φ maps bounded sets into bounded sets and for every
m ∈ N,
+
Φ(u) → −∞ as kuk → ∞, u ∈ X − ⊕ Xm
.
(2.7)
Then Φ has a nontrivial critical point.
3. Proofs of Theorems 1.1 and 1.2
It is reasonable to write the functional Φ in a form in which the quadratic part
is kuk2 . Let g(t) = f (t) + mt. By (F1)–(F2), it is known that G(t) satisfies the
following properties
t
m
G(t) ≤ g(t) + t2 ,
(3.1)
4
4
G(t)
lim
= +∞,
(3.2)
|t|→∞ t4
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QUASILINEAR SCHRÖDINGER EQUATIONS
5
and hence there is a Λ > 0 such that
G(t) ≥ −Λt4
for all t ∈ R.
(3.3)
Lemma 3.1. Suppose that (V1), (F1)–(F2) are satisfied, then Φ satisfies the (PS)
condition.
Proof. Suppose that {un } is a (PS) sequence. We claim that {un } is bounded. By
contradiction, we may assume that kun k → ∞. Using (2.3) and (3.1),we have
4 sup Φ(un ) + kun k ≥ 4Φ(un ) − (Φ0 (un ), un )
n
Z
2
= kun k +
g(un )un − 4G(un )dx
R
Z
≥ kun k2 − m u2n dx.
(3.4)
R
By (3.4), we obtain
kun k = O(|un |2 ).
(3.5)
Let vn = kun k un . Up to a subsequence, by the compact embedding X ,→ L2 (R)
we can assume that
−1
vn * v in X,
vn → v in L2 (R),
vn (x) → v(x) a.e in R.
By (3.5), we have
1
|un |2
= >0
c|un |2
c
for some positive constant c > 0. Therefor v 6= 0. Using (3.3), we obtain
Z
Z
Z
G(un )
G(un )
G(un )
dx
=
dx
+
dx
4
4
4
R kun k
v=0 kun k
v6=0 kun k
Z
Z
G(un ) 4
G(un ) 4
=
v
dx
+
v dx
n
4
u
u4n n
v=0
v6=0
n
Z
Z
G(un ) 4
≥ −Λ
vn4 dx +
v dx
u4n n
v=0
v6=0
= I1 + I2 .
|vn |2 ≥
Obviously, I1 ≥ −c > −∞. For x ∈ {x ∈ R|v 6= 0}, we have |un | → ∞. By (3.2),
we obtain
G(un ) 4
G(un )
=
v → +∞.
4
kun k
u4n n
By Fatou’s lemma, I2 → +∞. Then we have
Z
G(un )
dx → +∞.
(3.6)
4
R kun k
On the other hand
Z
Then we have
Z
1
2
2
G(un )dx = kun k +
u0n u2n dx − Φ(un )
2
R
R
1
2
≤ kun k + ckun k4 + C.
2
Z
G(un )dx = O(kun k4 ),
R
(3.7)
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Z. SHEN, Z. HAN
EJDE-2015/91
a contradiction to (3.6). So {un } is bounded.
Next, we show that such sequence {un } has a subsequence converging to a critical
point of Φ. Because {un } is bounded in X, we may assume un * u in X. Since
the imbedding X ,→ Lp (R) is compact, we have un → u in Lp (R). By a simple
computation, we have
Z
Z
Z
2
Φ0 (un )(un −u) = kun −uk2 +2 u0 (un −u)2 +2 u2 (u0 n −u0 )2 − g(un )(un −u).
R
R
R
By condition (F1) and Holder’s inequality, we have
Z
f (un )(un − u) ≤ C|u|p−1
|un − u|p .
p
R
Since un → u in Lp (R) and p ≥ 1, we have
Z
f (un )(un − u)dx → 0
as n → ∞.
(3.8)
as n → ∞.
(3.9)
R
Similarly,
Z
mun (un − u)dx → 0
R
By (3.8) and (3.9), we obtain
Z
g(un )(un − u) → 0
as n → ∞.
(3.10)
R
By the assumptions we have
Φ0 (un )(un − u) = o(kun − uk).
(3.11)
From (3.10) we obtain
Φ0 (un )(un − u) = kun − uk2 + 2
Z
2
u0 (un − u)2 + 2
R
Z
u2 (u0 n − u0 )2 + o(1).
R
Hence we obtain kun − uk → 0 as n → ∞. This completes the proof.
Lemma 3.2. Under assumptions (V1), (F1)–(F2), the functional Φ has a local
linking at 0 with respect to the decomposition X = X + ⊕ X − .
Proof. By (F1), there exists a C > 0 such that
|F (u)| ≤ C|u|p .
(3.12)
−
Using (2.1), (2.2), for u ∈ X , there exists a δ > 0. Then we have
Z
Z
Z
1
2
2
u0 + V (x)u2 dx + u0 u2 dx − F (u)dx
Φ(u) =
2 R
R
R
Z
1
02
2
4
p
u + V (x)u dx + Ckuk + C|u|p
≤
2 R
≤ −αkuk2 + Ckuk4 + C1 |u|pp
(3.13)
≤ −δkuk2 + Ckuk4 + C1 kukp .
If u ∈ X + , then there exists ξ > 0 such that
Z
Z
Z
1
02
2
02 2
Φ(u) =
u + V (x)u dx + u u dx − F (u)dx
2 R
R
R
≥ ξkuk2 − C1 kukp .
(3.14)
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QUASILINEAR SCHRÖDINGER EQUATIONS
7
By (3.13) and (3.14), there exists 0 < ρ < 1, such that
Φ(u) ≤ 0
for u ∈ X − , kuk ≤ ρ,
Φ(u) ≥ 0
for u ∈ X + , kuk ≤ ρ.
This completes the proof.
Lemma 3.3. Let Y be a finite dimensional subspace of X. Then Φ is anti-coercive
on Y ; that is,
Φ(u) → −∞ as kuk → ∞ u ∈ Y.
Proof. A similar lemma was proved in [5], we sketch the proof here for the reader’s
convenience. If the conclusion were not true, we can choose {un } ⊂ Y and ς ∈ R
such that
kun k → ∞, Φ(un ) ≥ ς.
(3.15)
Let vn = kun k−1 un . Since dim Y < ∞, up to a subsequence, we have
kvn − vk → 0,
vn (x) → v(x)
a.e. R
for some v ∈ Y with kvk = 1. If v(x) 6= 0, we have |un (x)| → ∞. Using (2.2) and
(3.6), we deduce
Z
1
G(un ) 4
Φ(un ) ≤ kun k
+
C
−
dx → −∞,
4
2kun k2
R kun k
a contradiction with (3.15).
Lemma 3.4. Suppose , (V1), (F1)-(F2) are satisfied. Then Φ satisfies condition
(C)∗ .
This proof is similar to the Lemma 3.1 and is omitted here. See also [5].
Proof of Theorem 1.1. It suffices to verify that
(i) bk = inf u∈Zk ,kuk=rk Φ(u) → +∞ as k → ∞,
(ii) αk = maxu∈Yk ,kuk=ρk Φ(u) ≤ 0.
(i) We claim that for any 2 ≤ p, we have
βk :=
kukLp → 0,
sup
as k → ∞.
(3.16)
u∈Zk ,kuk=1
If the conclusion were not true, we may assume that βk → β > 0 as k → ∞. Then
there exists a uk ∈ Zk with kuk k = 1 and kuk kp ≥ β2 for large k. By the Parseval
equality we have
∞
X
hu, uk i = h
αj φj , uk i
j=k
≤k
=
∞
X
j=k
∞
X
j=k
αj φj k kuk k
αj2
1/2
→0
8
Z. SHEN, Z. HAN
EJDE-2015/91
where h, i denotes the inner product in X. Using the Riesz-Frechet representation
theorem, we obtain that uk * 0 and thus uk → 0 in Lp . This is a contradiction.
For u ∈ Zk with kuk = rk , for enough small ,
Z
Z
Z
1
02
2
02 2
Φ(u) =
u + V (x)u dx + u u dx − F (u)dx
2 R
R
R
Z
2
≥ kkuk −
F (u)dx
R
2
≥ kkuk − C|u|pp
≥ kkuk2 − Cβkp kukp .
Choosing rk = βk−1 , we have
Φ(u) ≥ kβk−2 − C → +∞.
This proves (i).
(ii) Since dim Yk < ∞, using Lemma 3.3, we have
Φ(u) → −∞ for u ∈ Yk and ρk → ∞.
Then we obtain
αk =
max
Φ(u) ≤ 0.
u∈YK ,kuk=ρk
This completes the proof.
Proof of Theorem 1.2. In Lemmas 3.2 and 3.4, we see that Φ satisfies condition
+
) < ∞, By Lemma 3.3, we
(C)∗ , and has a local linking at 0. Since dim(X − ⊕ Xm
−
+
have Φ(u) → −∞, as kuk → ∞, u ∈ X ⊕ Xm . By Theorem 2.3, equation (1.1)
has at least one nontrivial solution.
Acknowledgments. This research was supported by NSFC (11171047), NSFC
(11171204) and GDNSF (S2012010010038).
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Zupei Shen
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024,
China
E-mail address: pershen@mail.dlut.edu.cn
Zhiqing Han (corresponding author)
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024,
China
E-mail address: hanzhiq@dlut.edu.cn Phone(Fax) +86 41184707268