Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 98, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
INFINITELY MANY SOLUTIONS FOR SUBLINEAR
KIRCHHOFF EQUATIONS IN RN WITH SIGN-CHANGING
POTENTIALS
ANOUAR BAHROUNI
Abstract. In this article we study the Kirchhoff equation
Z
“
”
− a+b
|∇u|2 dx ∆u + V (x)u = K(x)|u|q−1 u,
RN
in RN ,
where N ≥ 3, 0 < q < 1, a, b > 0 are constants and K(x), V (x) both change
sign in RN . Under appropriate assumptions on V (x) and K(x), the existence
of infinitely many solutions is proved by using the symmetric Mountain Pass
Theorem.
1. Introduction
In this article, we study the existence of infinitely many solutions of the nonlinear
Kirchhoff equation
Z
− a+b
|∇u|2 dx ∆u + V (x)u = K(x)|u|q−1 u, in RN ,
(1.1)
RN
1
N
q+1
N
u ∈ H (R ) ∩ L (R ),
where N ≥ 3, 0 < q < 1, a, b are positive constants and K(x), V (x) ∈ L∞ (RN )
both change sign in RN and satisfies some conditions specified below.
If in (1.1), we set V (x) = 0 and replace RN by a bounded domain Ω ⊂ RN , then
(1.1) reduces to the problem
Z
− a+b
|∇u|2 dx ∆u = f (x, u) in Ω,
(1.2)
Ω
u = 0 on ∂Ω,
where f (x, u) ∈ C(RN × R, R). Problem (1.2) is related to the stationary analogue
of the Kirchhoff equation
Z
utt − a + b
|∇x u|2 dx ∆x u = f (x, u)
Ω
2000 Mathematics Subject Classification. 35J60, 35J91, 58E30.
Key words and phrases. Kirchhoff equations; symmetric Mountain Pass Theorem;
infinitely many solutions.
c
2013
Texas State University - San Marcos.
Submitted March 6, 2013, Published April 16, 2013.
1
2
A. BAHROUNI
EJDE-2013/98
which was proposed by Kirchhoff in 1883 [14] as a generalization of the well-known
d’Alembert’s wave equation
Z L
∂2u
∂u
∂ 2 u P0
E
| |2 dx 2 = f (x, u)
ρ 2 −
+
∂ t
h
2L 0 ∂x
∂ x
for free vibrations of elastic strings. Kirchhoff’s model takes into account the
changes in length of the string produced by transverse vibrations. Some early
classical investigations of Kirchhoff equations can be seen in Bernstein [5] and
Pohoẑaev [18]. Problem (1.1) called the attention of several researchers mainly
after the work of Lions [15], where a functional analysis approach was proposed.
Recently, problems like type (1.1) have been investigated by several authors; see
[1, 2, 7, 9, 10, 11, 12, 16, 17, 19].
Ma and Rivera [16] obtained a positive solutions of (1.2) by using variational
methods. He and Zou [11] showed that problem (1.2) admits infinitely many solutions by using the local minimum methods and the Fountain Theorem. Very
recently, some authors have studied the Kirchhoff equation on the whole space
RN . Since (1.1) is set on RN , it is well known that the Sobolev embedding
H 1 (RN ) ,→ Lm (RN ) (2 ≤ m ≤ 2∗ = N2N
−2 ) is not compact and then it is usually difficult to prove that a minimizing sequence or a Palais-Smale sequence is
strongly convergent if we seek solution of (1.1) by variational methods. If V (x) is
radial, as in [17, 19] we can avoid the lack of compactness of Sobolev embedding by
looking for solution of (1.1) in the subspace of radial functions of D1,2 (RN ) since
the embedding is compact. Nie and Wu [17] studied a Schrödinger-Kirchhoff-type
equation with radial potential and they proved the existence of infinitely many solutions by using a symmetric Mountain Pass Theorem. On the other hand, Alves
and Figueiredo [1] studied a periodic Kirchhoff equation in RN . They proved the
existence of a nontrivial solution when the nonlinearity is in subcritical and critical
case. Liu and He [12] proved the existence of infinitely many high-energy solutions
where V (x) ≥ 0 and the nonlinearity is superlinear. We remark that when a = 1
and b = 0, problem (1.1) can be rewritten as the well-known Schrödinger equation
− ∆u + V (x)u = K(x)|u|q−1 u,
x ∈ RN .
(1.3)
For this equation, there is a large body of literature on the existence and multiplicity
of solutions; we for example [4, 6, 8, 13, 20] and the references therein. Motivated
by the above fact, in this paper our aim is to study the existence of infinitely many
nontrivial solutions for (1.1) when 0 < q < 1 and K(x), V (x) both change sign.
Our tool is the symmetric Mountain Pass Theorem.
To state our main result we require the following assumptions:
(A1) V ∈ L∞ (RN ) and there exist β, R0 > 0 such that V (x) ≥ β for all |x| ≥ R0 ;
(A2) K ∈ L∞ (RN ) and there exist α, R1 , R2 > 0, y0 = (y1 , · · · , yN ) ∈ RN such
that K(x) ≤ −α for all |x| ≥ R1 and K(x) > 0 for all x ∈ B(y0 , R2 ).
Our main result reads as follows.
Theorem 1.1. Under assumptions (A1), (A2), problem (1.1) admits infinitely
many nontrivial solutions.
In the next section we give some notation and preliminary results; and in section
3, we prove theorem (1.1).
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SUBLINEAR KIRCHHOFF EQUATIONS
3
2. Preliminaries
We will us the following notation: Let
Z
1/m
kukm =
|u(x)|m dx
,
1 ≤ m < +∞.
RN
Let 2∗ = N2N
−2 for all N ≥ 3. Let BR denote the ball centred in zero of radius R > 0
c
in RN and BR
= RN \BR . Let F 0 (u) : the Fréchet derivative of F at u. For s, be
the Sobolev constant in
kuk2∗ ≤ sk∇uk2 ,
∀u ∈ H 1 (RN ).
Let E = H 1 (RN ) ∩ Lq+1 (RN ), 0 < q < 1, endowed with the norm
kuk = k∇uk2 + kukq+1 ,
The space E becomes a reflexive Banach space.
Problem (1.1) has a variational structure. Indeed we consider the functional
I : E → R defined by
Z
Z
Z
2 a Z
1
1
b
|∇u|2 dx +
|∇u|2 dx+
V (x)u2 dx−
K(x)|u|q+1 dx.
I(u) =
4 RN
2 RN
2 RN
q + 1 RN
As is well known, I is of class C 1 on E and any critical point of I is a solution of
(1.1).
A functional I is said to satisfy the Palais-Smale condition (PS, for short) if for
very sequence (un ) such that
I(un ) is bounded, and kI 0 (un )k → 0,
there is a convergent subsequence of (un ).
Before proving Theorem (1.1), we give the symmetric Mountain Pass Theorem.
Definition 2.1. Let E be a Banach space and A a subset of E. Set A is said to
be symmetric if u ∈ E implies −u ∈ E. For a closed symmetric set A which does
not contain the origin, we define a genus γ(A) of A by the smallest integer k such
that there exists an odd continuous mapping from A to Rk \{0}. If there does not
exist such a k, we define γ(A) = ∞. We set γ(∅) = 0. Let Γk denote the family of
closed symmetric subsets A of E such that 0 ∈
/ A and γ(A) ≥ k.
Now we give the symmetric Mountain Pass Theorem [2] which improved by
Kajikiya [13] to obtain the following Theorem.
Theorem 2.2. Let E be an infinite dimensional Banach space and I ∈ C 1 (E, R)
satisfy:
(1) I is even, bounded from below, I(0) = 0 and I satisfies the Palais-Smale
condition.
(2) For each k ∈ N, there exists an Ak ∈ Γk such that
sup I(u) < 0.
u∈Ak
Then either of the following tow conditions holds:
(i) There exists a sequence (uk ) such that I 0 (uk ) = 0, I(uk ) < 0 and (uk )
converges to zero; or
(ii) There exist two sequences (uk ) and (vk ) such that I 0 (uk ) = 0, I(uk ) = 0,
uk 6= 0, limk→+∞ uk = 0, I 0 (vk ) = 0, I(vk ) < 0, limk→+∞ I(vk ) = 0 and
(vk ) converges to a non-zero limit.
4
A. BAHROUNI
EJDE-2013/98
3. Proof of Theorem (1.1)
Lemma 3.1. Under assumptions (A1), (A2), the functional I is bounded from
below.
Proof. By (A1), (A2) and Hölder inequality, we have
Z
Z
Z
2 a Z
b
1
1
2
2
2
I(u) =
|∇u| dx +
|∇u| dx +
V (x)u dx −
K(x)|u|q+1 dx
4 RN
2 RN
2 RN
q + 1 RN
Z
Z
Z
1
1
b
|∇u|2 dx)2 −
V − (x)u2 dx −
K + (x)|u|q+1 dx
≥ (
4 RN
2 RN
q + 1 RN
b
s2
≥ k∇uk42 − kV − kN/2 k∇uk22 − sq+1 kK + k ∗ 2∗ k∇ukq+1
.
2
2 −q−1
4
2
Since 0 < q < 1, we conclude the proof.
Lemma 3.2. Assume (A1), (A2) hold. Then, any (P S) sequence (un ) of I is
bounded in E.
Proof. Let (un ) be a (PS) sequence of I. Then, there exists a positive constant
c > 0 such that
c ≥ I(un )
Z
Z
2 a Z
b
1
=
|∇un |2 dx +
|∇un |2 dx +
V (x)u2n dx
4 RN
2 RN
2 RN
Z
1
−
K(x)|un |q+1 dx
q + 1 RN
Z
Z
Z
1
1
b
|∇un |2 dx)2 −
V − (x)u2n dx −
K + (x)|un |q+1 dx
≥ (
4 RN
2 RN
q + 1 RN
b
s2
≥ k∇un k42 − kV − kN/2 k∇un k22 − sq+1 kK + k ∗ 2∗ k∇un kq+1
.
2
2 −q−1
4
2
Hence, there exists γ0 > 0 such that
k∇un k2 ≤ γ0 ,
∀n ∈ N.
(3.1)
On the other hand, there exists c > 0 such that
c+
kun k
1
≥ − hI 0 (un ), un i + I(un )
4
4Z
Z
Z
a
1
1
1
=
|∇un |2 dx +
V (x)u2n dx + ( −
)
K(x)|un |q+1 dx
4 RN
4 RN
4 q + 1 RN
Z
1
1
≥(
− )
(K − (x) + χBR1 (x))|un |q+1 dx
q + 1 4 RN
Z
Z
1
1
1
+
q+1
−(
− )
(K (x) + χBR1 (x))|un | dx −
V − (x)u2n dx
q + 1 4 RN
4 RN
Z
s2 kV − kN/2
1
1
≥(
− ) min(α, 1)
|un |q+1 (x)dx −
k∇un k2
q+1 4
4
RN
1
1
− sq+1 (
− )kK + + χBR1 k ∗ 2∗ k∇un kq+1
.
2
2 −q−1
q+1 4
Therefore, by using (3.1), we obtain
kun kq+1 ≤ γ1 ,
for some γ1 > 0.
(3.2)
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SUBLINEAR KIRCHHOFF EQUATIONS
Combining (3.1) and (3.2), we conclude the proof.
5
We need the following Lemma to prove that the Palais-Smale condition is satisfied for I on E.
Lemma 3.3. Let x and y two arbitrary real numbers, then there exists a constant
c > 0 such that
|x + y|q+1 − |x|q+1 − |y|q+1 ≤ c|x|q y
(3.3)
Proof. If x = 0, the inequality (3.3) is trivial. Suppose that x 6= 0. We consider
the continuous function f defined on R\{0} by
f (t) =
|1 + t|q+1 − |t|q+1 − 1
.
|t|
Note that lim|t|→+∞ f (t) = 0 and limt→0± f (t) = ±(q + 1). Then there exists a
constant c > 0 such that |f (t)| ≤ c, for all t ∈ R\ {0}. In particular |f ( xy )| ≤ c, so
|1 + y |q+1 − | y |q+1 − 1 ≤ c| y |,
x
x
x
multiplying by |x|q+1 , we obtain the desired result.
Lemma 3.4. Assume that (a1), (A2) hold. Then I satisfies the Palais-Smale
condition in E.
Proof. Let (un ) be a (P S) sequence. By Lemma (3.2), (un ) is bounded in E. Then
there exists a subsequence un * u in E, un → u in LpLoc (RN ) for all 1 ≤ p ≤ 2∗
and un → u a.e in RN .
By [8], it is sufficient to prove that for any > 0, there exist R3 > 0 and n0 ∈ N∗
such that
Z
(|∇un |2 + |un |q+1 )dx ≤ , for all R ≥ R3 and n ≥ n0 .
|x|≥R3
c
Let φR be a cut-off function so that φR = 0 on B R , φR = 1 on BR
, 0 < φR < 1
2
and
c
|∇φR |(x) ≤ , for all x ∈ RN .
(3.4)
R
We can easily remark that for any u ∈ E and R ≥ 1,
kφR uk ≤ ckuk.
(3.5)
Since I 0 (un ) → 0 in E 0 as n → +∞, we know that for any > 0, there exists n0 > 0
such that
|hI 0 (un ), φR un i| ≤ ckI 0 (un )kE 0 kun k ≤ , ∀n ≥ n0 ;
3
that is, n ≥ n0 . Then
Z
Z
(a + b
|∇un |2 dx)
|∇un |2 φR (x)dx
RN
RN
Z
Z
2
+
V (x)|un | φR (x)dx −
K(x)|un |q+1 φR (x)dx ≤ .
3
N
N
R
R
6
A. BAHROUNI
EJDE-2013/98
Hence,
Z
(a|∇un |2 + (K − + χBR1 )(x)|un |q+1 )φR (x)dx
Z
Z
≤
V − (x)u2n φR dx − a
un ∇un ∇φR dx
RN
RN
Z
+
(K + + χBR1 )(x)|un |q+1 φR dx + .
3
RN
By Hölder inequality and (3.4), there exists R4 > 0 such that
Z
c
un ∇un ∇φR dx ≤
< , ∀|x| ≥ R4 .
R
3
RN
RN
(3.6)
(3.7)
From (A1) and (A2), there exists R5 > 0 such that
Z
Z
V − (x)u2n φR dx +
(K + + χBR1 )(x)|un |q+1 dx
RN
RN
≤ ckV − φR kN/2 + ck(K + + χBR1 )φR k ∗ 2∗
2 −q−1
≤
for |x| ≥ R5 .
3
Put R3 = max(R4 , R5 ). By (3.6), (3.7) and (3.8), we have
Z
min(a, min(α, 1))
(|∇un |2 + |un |q+1 )φR dx ≤ .
(3.8)
RN
The proof is complete.
Lemma 3.5. Assume (A1), (A2) hold. Then for each k ∈ N, there exists an
Ak ∈ Γk such that
sup I(u) < 0.
u∈Ak
Proof. We use the following geometric construction introduced by Kajikiya [13]:
Let R2 and y0 be fixed by assumption (A1) and consider the cube
D(R2 ) = {(x1 , · · · , xN ) ∈ RN : |xi − yi | < R2 , 1 ≤ i ≤ N }.
Fix k ∈ N arbitrarily. Let n ∈ N be the smallest integer such that nN ≥ k. We
divide D(R2 ) equally into nN small cubes, denote them by Di with 1 ≤ i ≤ nN ,
by planes parallel to each face of D(R2 ). The edge of Di has the length of z = Rn2 .
We construct a new cubes Ei in Di such that Ei has the same center as that of Di .
The faces of Ei and Di are parallel and the edge of Ei has the length of z2 . Then,
we make a function ψi , 1 ≤ i ≤ k, such that
supp(ψi ) ⊂ Di ,
ψi (x) = 1
supp(ψi ) ∩ supp(ψj ) = ∅
for x ∈ Ei ,
0 ≤ ψi (x) ≤ 1,
(i 6= j),
∀x ∈ RN .
We denote
S k−1 = {(t1 , · · · , tk ) ∈ Rk : max |ti | = 1},
1≤i≤k
k
X
Wk = {
ti ψi (x) : (t1 , · · · , tk ) ∈ S k−1 } ⊂ E.
i=1
(3.9)
(3.10)
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SUBLINEAR KIRCHHOFF EQUATIONS
7
Pk
Since the mapping (t1 , . . . , tk ) → i=1 ti ψi from S k−1 to Wk is odd and homeomorphic, then γ(Wk ) = γ(S k−1 ) = k. Wk is compact in E, then there is a constant
αk > 0 such that
kuk2 ≤ αk for all u ∈ Wk .
we need to recall the inequality
kuk2 ≤ ck∇ukr2 kuk1−r
q+1 ≤ ckuk
with r =
∗
2 (q−1)
2(2∗ −q−1) .
Then, there is a constant ck > 0 such that
kuk22 ≤ ck
Let z > 0 and u =
Pk
i=1 ti ψi (x)
for all u ∈ Wk .
∈ Wk ,
k
I(zu) ≤ z 4 b
(3.11)
αk2
az 2
kV k∞
1 X
+
αk + z 2
ck −
4
2
2
q + 1 i=1
Z
K(x)|zti ψi |q+1 dx. (3.12)
Di
By (3.9), there exists j ∈ [1, k] such that |tj | = 1 and |ti | ≤ 1 for i 6= j. Then
k Z
X
K(x)|zti ψi |q+1 dx
Di
i=1
Z
=
K(x)|ztj ψj |q+1 dx +
Z
K(x)|ztj ψj |q+1 dx +
Dj \Ej
Ej
XZ
i6=j
K(x)|zti ψi |q+1 dx .
Di
(3.13)
Since ψj (x) = 1 for x ∈ Ej and |tj | = 1, we have
Z
Z
K(x)|ztj ψj |q+1 dx = |z|q+1
K(x)dx.
(3.14)
K(x)|zti ψi |q+1 dx ≥ 0.
(3.15)
Ej
Ej
On the other hand by (A1) we obtain
Z
XZ
K(x)|ztj ψj |q+1 dx +
Dj \Ej
i6=j
Di
From (3.12), (3.13), (3.14) and (3.15), we obtain
2
I(zu)
aαk
kV k∞
|z|q+1
2 bαk
≤
z
+
+
c
−
inf
k
z2
4
2
2
z 2 1≤i≤k
Z
K(x)dx .
Ei
It follows that
lim sup
z→0 u∈Wk
I(zu)
= −∞.
z2
We fix z small such that
sup{I(u), u ∈ Ak } < 0,
where Ak = zWk ∈ Γk .
The proof is complete.
Proof of Theorem (1.1). Evidently, I(0) = 0 and I is an odd functional. Then by
Lemmas (3.1), (3.4) and (3.5), conditions (1) and (2) of Theorem 2.2 are satisfied.
Then, by Theorem 2.2, problem (1.1) admits an infinitely many solutions (uk ) ∈ E
which converging to 0 and u0 can be supposed nonnegative since
I(u0 ) = I(|u0 |).
8
A. BAHROUNI
EJDE-2013/98
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Anouar Bahrouni
Mathematics Department, University of Monastir, Faculty of Sciences, 5019 Monastir,
Tunisia
E-mail address: bahrounianouar@yahoo.fr, Fax + 216 73 500 278