Electronic Journal of Qualitative Theory of Differential Equations
2017, No. 32, 1–14; doi: 10.14232/ejqtde.2017.1.32
http://www.math.u-szeged.hu/ejqtde/
Existence of sign-changing solution with least energy
for a class of Kirchhoff-type equations in R N
Xianzhong Yao B 1, 2 and Chunlai Mu2
1 Faculty
of Applied Mathematics, Shanxi University of Finance and Economics
Taiyuan 030006, P.R. China
2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China
Received 21 January 2017, appeared 9 May 2017
Communicated by Dimitri Mugnai
Abstract. We consider the existence of least energy sign-changing (nodal) solution
of Kirchhoff-type elliptic problems with general nonlinearity. Using a truncated technique and constrained minimization on the nodal Nehari manifold, we obtain that the
Kirchhoff-type elliptic problem possesses one least energy sign-changing solution by
applying a Pohožaev type identity. Moreover, the energy of the sign-changing solution
is strictly more than the ground state energy.
Keywords: Kirchhoff-type, ground state solution, sign-changing solution, Pohožaev
type identity.
2010 Mathematics Subject Classification: 35J50, 35B38.
1
Introduction
In this paper, we are concerned with the following Kirchhoff-type elliptic problem with general nonlinearity:
Z
Z
2
2
a+λ
|∇u| dx + λb
u dx [−∆u + bu] = f (u), x ∈ R N ,
(1.1)
RN
RN
where a, b > 0 are constants, λ > 0 is a parameter and N ≥ 3. Moreover, f ∈ C 1 (R, R+ )
satisfies the following hypotheses:
( f 1 ) | f (t)| ≤ C (|t| + |t|q−1 ) for q ∈ (2, 2∗ ), 2∗ =
( f 2 ) f (t) = o (|t|) as t → 0;
f (t)
t→∞ |t|
( f 3 ) lim
( f4 )
f (t)
|t|
= +∞;
is strictly increasing in R\{0}.
B Corresponding
author. Email: [email protected]
2N
N −2 ;
X. Yao and C. Mu
2
Kirchhoff-type problems are often referred to as being nonlocal because of the presence
of the integral terms. It is related to the stationary analogue of the equation that arise in the
study of string or membrane vibrations, namely
∂2 u
ρ 2 −
∂t
P0
E
+
h
2L
2
Z L
∂u 2
∂ u
| | dx
= 0,
2
0
∂x
∂x
(1.2)
which was presented by Kirchhoff [10] in 1883. This model is an extension of the classical
d’Alembert wave equation by considering the effects of the changes on the length of the
elastic string during the free vibrations. The parameters in the Kirchhoff’s model have the
following meanings: L is the length of the string, h is the area of cross-section, E is the Young
modulus of the material, ρ is the mass density and P0 is the initial tension. Some early classical
studies of Kirchhoff-type equations were those of Pohožaev [22] and Bernstein [3]. However,
Kirchhoff’s model received great attention only after Lions [13] proposed following abstract
framework for the model (1.2),
(
utt − ( a + b
R
Ω
|∇u|2 dx )∆u = f ( x, u),
x ∈ Ω,
x ∈ ∂Ω.
u=0
(1.3)
The existence and concentration behavior of solutions to Kirchhoff-type elliptic problem
have been extensively studied in the past decade. Most researchers paid their attention to focus on existence of positive solutions, ground state, radial and nonradial solutions and semiclassical state under some different assumptions, see for example [1,4,6,7,11,12,17,19–21,24,26]
and references therein. While existence of sign-changing solutions has been received few attention, and there are very few results on existence of sign-changing solutions to Kirchhofftype problem. Only Zhang et al. [18, 28] investigated the existence of sign-changing solution
of the Kirchhoff-type problem (1.4),
(
−( a + b
R
Ω
|∇u|2 dx )∆u = f (u),
x ∈ Ω,
x ∈ ∂Ω,
u=0
(1.4)
where a > 0, b ≥ 0 and Ω ⊂ R N ( N ≥ 1) is a bounded domain with smooth boundary. By
using variational methods and invariant sets of descent flow, they demonstrated that equations (1.4) possesses a sign-changing solution with nonlinearity f satisfying some suitable
conditions.
In recent years, there has been increasing attention to the existence of sign-changing
(nodal) solutions to Kirchhoff-type problem. In [23], Shuai considered equations (1.4) in
N = 1, 2, 3 with f ∈ C 1 (R, R) satisfying following conditions:
( H1 ) f (t) = o (|t|) as t → 0;
( H2 ) for some constant p ∈ (4, 2∗ ), lim tfp(−t)1 = 0, where 2∗ = +∞ for N = 1, 2, if N = 3,
t→∞
2∗ = 6;
F (t)
4
t→∞ t
( H3 ) lim
( H4 )
f (t)
| t |3
= +∞, where F (t) =
Rt
0
f (s)ds;
is an increasing function in R\{0}.
Existence of sign-changing solution of Kirchhoff-type equations
3
Employing constraint variational method and quantitative deformation lemma, author asserted that there is one least energy sign-changing solution (nodal solution), which has precisely two nodal domains. Moreover, the energy of sign-changing solution is strictly larger
than the ground state energy. While Figueiredo and Nascimento in [5] discussed the following
more general problem than (1.4), for N = 3,
(
R
−M
2 dx ∆u = f ( u ),
|∇
u
|
Ω
x ∈ Ω,
x ∈ ∂Ω,
u=0
(1.5)
where M, f ∈ C 1 (R, R) fulfill some assumptions:
( M1 ) function M is increasing and M(0) := m0 > 0;
( M2 )
M(t)
t
is a decreasing function for t > 0;
e3 ) there is θ ∈ (4, 6) such that 0 < θF (t) ≤ f (t)t, for t 6= 0.
(H
e3 ), ( H4 ), they explored that there
Under the conditions ( M1 ), ( M2 ) and ( H1 ), ( H2 ), ( H
exists one least energy nodal solution to the problem (1.5). For more results, we refer to
[2, 16, 27] for some variant version of Kirchhoff-type problem.
From the discussion above, we discover that researchers usually need suppose that f sate3 ), which ensure the boundedness of a minimum sequence for
isfies ( H4 ) and ( H3 ) or ( H
the corresponding functional of the Kirchhoff-type problem. As well it also guarantees that
the nodal Nehari manifold of corresponding functional of the Kirchhoff-type problem is not
empty. Then their results can be derived by usual variational methods and quantitative dee3 ) by the
formation lemma. In this paper, we replace the conditions ( H4 ) and ( H3 ) or ( H
hypotheses ( f 4 ) and ( f 3 ), which is weaker than the conditions in foregoing literatures. A
typical case is that f (u) = |u| p−1 u for p ∈ (1, 5), however, the results in the references above
is valid only for p ∈ (3, 5). To the best authors’ knowledge, there is no result on the existence
of least energy sign-changing (nodal) solution to Kirchhoff-type problem with nonlinearity f
satisfying the hypotheses ( f 3 ) and ( f 4 ).
To character our results, we need first to introduce the energy functional for corresponding
Kirchhoff-type problem (1.1) and nodal Nehari manifold. Let H 1 (R N ) be the usual Sobolev
space equipped with the inner product and norm
(u, v) =
Z
R3
∇u∇v + buvdx,
kuk = (u, u)1/2 ,
and L p (R N ) is the usual Lebesgue space endowed with the norm
|u| p =
as well as
Z
RN
|u| p dx
1/p
,
for 1 ≤ p < ∞,
|u|∞ = sup |u( x )|,
x ∈R N
∗
D 1,2 (R N ) := {u ∈ L2 (R N ) : ∇u ∈ L2 (R N )}
with norm kukD1,2 (RN ) = |∇u|2 . It is well known that the embedding of H 1 (R N ) into L p (R N )
for p ∈ [2, 2∗ ] is continuous but not compact. Denote the subspace Hr1 (R N ) := {u ∈
H 1 (R N ) : u is radial symmetric function} and hereafter, for simplicity, H := Hr1 (R N ). Then
H ,→ L p (R N ) compactly for p ∈ (2, 2∗ ), see [25, Corollary 1.26].
X. Yao and C. Mu
4
Define the energy functional associated with equation (1.1), Jλ : H → R given by
Jλ (u) =
a
λ
k u k2 + k u k4 −
2
4
Z
RN
F (u)dx.
Obviously, Jλ belong to C 1 ( H, R). For any u, v ∈ H, there is
h Jλ0 (u), vi
2
= a(u, v) + λkuk (u, v) −
Z
RN
f (u)vdx.
It is well-known that each weak solution of equation (1.1) corresponds a critical point of Jλ .
We define the Nehari manifold for the corresponding energy functional Jλ
Nλ = {u ∈ H \{0} : h Jλ0 (u), ui = 0},
and the nodal Nehari manifold
Mλ = {u ∈ H : u± 6= 0, h Jλ0 (u), u± i = 0},
where
u+ ( x ) = max{u( x ), 0}
and
u− ( x ) = min{u( x ), 0}.
ceλ := inf{ Jλ (u) : u ∈ Nλ }
and
cλ := inf{ Jλ (u) : u ∈ Mλ }.
Moreover, denote
When u is a nontrivial solution to equation (1.1) and Jλ (u) ≤ Jλ (v), where v is any solution
of equation (1.1), then we say that u ∈ H is a ground state (least energy) solution to equation
(1.1) and u is one sign-changing (nodal) solution to equation (1.1) if u± 6= 0. By Lemma 2.3
below, we have that Nλ and Mλ are not empty and Mλ ⊂ Nλ . From the definition of Nλ and
Mλ , we know that all nontrivial solutions and sign-changing solutions to equation (1.1) are
included in Nλ and Mλ , respectively.
Now, we give our main results as follows.
Theorem 1.1. Assume the conditions ( f 1 )–( f 4 ) hold. Then there exists a positive Λ such that, for any
λ ∈ (0, Λ), the problem (1.1) have a ground state solution uλ which is constant sign and a least energy
sign-changing solution vλ satisfying
cλ = Jλ (vλ ) > Jλ (uλ ) = ceλ > 0.
The remainder of this paper is organized as follows. In Section 2, we present the abstract
framework of the problem as well as some preliminary results. Theorem 1.1 will be proved in
Section 3.
2
Preliminaries
In this section, we show examples how theorems, definitions, lists and formulae should be
formatted.
In this section, we give some notations and lemmas. According to the foregoing discussion,
we know that it is very difficult to obtain bounded minimum sequences for the associated
Existence of sign-changing solution of Kirchhoff-type equations
5
functional Jλ . So we here use a truncated technique, following [8, 9, 11], to handle it. We
introduce a cut-off function φ ∈ C ∞ (R, R) satisfying


φ(t) = 1,
t ∈ [0, 1],



0 ≤ φ(t) ≤ 1, t ∈ (1, 2),
t ∈ [2, ∞),

φ(t) = 0,



 0
|φ |∞ ≤ 2,
and then consider the following truncated functional Jλ,κ : H → R defined by
Jλ,κ (u) =
λ
a
kuk2 + hκ (u)kuk4 −
2
4
Z
RN
F (u)dx,
where for every κ > 0,
hκ ( u ) = φ
k u k2
κ2
.
It is easy to know that Jλ,κ belong to C 1 ( H, R). For κ > 0 enough large, we can take advantage
of Jλ,κ to obtain a critical point wλ of Jλ,κ , then, by the definition of φ and Jλ,κ , we know that
wλ is a critical point of Jλ if we show that kwλ k ≤ κ. We define the Nehari manifold of Jλ,κ as
follows
0
Nλ,κ = {u ∈ H \{0} : h Jλ,κ
( u ), u i = 0}
and the nodal Nehari manifold
0
( u ), u ± i = 0}.
Mλ,κ = {u ∈ H : u± 6= 0, h Jλ,κ
Moreover, denote
ceλ,κ := inf{ Jλ,κ (u) : u ∈ Nλ,κ },
cλ,κ := inf{ Jλ,κ (u) : u ∈ Mλ,κ }.
Notation 2.1. Throughout this paper, we denote by “→” and “*” the strong and weak convergence in the related function space, respectively. Br ( x ) := {y ∈ R N : | x − y| < r }. We use
o (1) to denote any quantity which tends to zero as n → ∞. We will use the symbol C and Ci
for denoting positive constants unless otherwise stated explicitly and the value of C and Ci is
allowed to change from line to line and also in the same formula.
Lemma 2.2. For all u ∈ Nλ,κ , the following results hold:
(i ) for any λ > 0, There exists r > 0 such that kuk ≥ r;
(ii ) Jλ,κ has a lower bound in Nλ,κ .
Proof. For any u ∈ Nλ,κ , there is
λ
akuk + λhκ (u)kuk + 2 φ0
2κ
2
4
k u k2
κ2
k u k6 =
Z
RN
f (u)udx,
(2.1)
By ( f 1 ), ( f 2 ) and Sobolev’s inequality, it is easy to obtain the result (i ) if kuk2 ≥ 2κ 2 , otherwise,
the following inequality holds
k u k2
λ
λ
akuk2 + λhκ (u)kuk4 + 2 φ0
k u k6 ≥ a k u k2 − 2 k u k6 ,
2κ
κ2
κ
X. Yao and C. Mu
6
owing to ( f 1 ) and ( f 2 ), we have, for small ε > 0,
Z
q
RN
f (u)udx ≤ ε|u|22 + Cε |u|q .
(2.2)
Combining the three formulas above and Sobolev inequality, we obtain that
a k u k2 −
λ
k u k6 ≤
κ2
Z
q
RN
f (u)udx ≤ ε|u|22 + Cε |u|q ≤ εC1 kuk2 + C2 kukq .
It follows the assertion (i ).
Next we show the item (ii ). If kuk2 ≥ 2κ 2 for all u ∈ N , by the definition of φ, we observe
Jλ,κ (u) =
a
k u k2 −
2
Z
RN
F (u)dx,
and by (2.1), it holds
a k u k2 =
Z
RN
f (u)udx.
Since ( f 4 ) implies that 2F (t) ≤ f (t)t for t ∈ R, we deduce that Jλ,κ (u) > 0 and the result is
finished. Suppose, by contradiction, that there is u ∈ N such that kuk2 < 2κ 2 . In which case,
the result is valid by Jλ,κ ∈ C 1 ( H, R). Thus the conclusion is established.
Lemma 2.3. For any u ∈ H with u± 6= 0, then there is a pair (tu , su ) ∈ R+ × R+ such that
tu u+ + su u− ∈ Mλ,κ for λ small. In particular, Mλ,κ 6= ∅ and for all (t, s) ∈ R+ × R+ , there is
Jλ,κ (tu u+ + su u− ) ≥ Jλ,κ (tu+ + su− ).
Proof. For any u ∈ H with u± 6= 0, define function g : [0, ∞) × [0, ∞) → R given by
g(t, s) := Jλ,κ (tu+ + su− )
and its gradient Φ : [0, ∞) × [0, ∞) → R × R, denoted by
∂g
∂g
Φ(t, s) := Φ1 (t, s), Φ2 (t, s) =
(t, s), (t, s)
∂t
∂s
0
+
−
+
0
+
= h Jλ,κ (tu + su ), u i, h Jλ,κ (tu + su− ), u− i .
We simply compute, by ( f 1 )( f 2 ) and Sobolev inequality,
at2 + 2
as2 − 2
q
q
ku k − εt2 |u+ |22 − Ctq |u+ |q +
ku k − εs2 |u− |22 − Csq |u− |q
2
2
at2 + 2
as2 − 2
≥
ku k − εC1 t2 ku+ k2 − C2 tq ku+ kq +
ku k − εC3 s2 ku− k2 − C4 sq ku− kq ,
2
2
for small ε > 0 and some positive constants Ci (i = 1, 2, 3, 4). Therefore, g(t, s) is positive for
(t, s) small. Since ( f 3 ), for t large enough, there exists a large M > 0 such that
g(t, s) ≥
f ( t ) ≥ M | t |.
(2.3)
Thus, for (t, s) large enough, we compute
g(t, s) = Jλ,κ (tu+ + su− )
Z
a 2 + 2 a 2 − 2 λ
+
−
+
− 4
= t ku k + s ku k + hκ (tu + su )ktu + su k −
F (tu+ + su− )dx
2
2
4
RN
Z
(2.4)
a
a
= t2 k u + k2 + s2 k u − k2 −
F (tu+ ) + F (su− )dx
2
2
RN
Z
Z
a 2 + 2 a 2 − 2
≤ t ku k + s ku k − Mt2
|u+ |2 dx − Ms2
|u− |2 dx,
2
2
RN
RN
Existence of sign-changing solution of Kirchhoff-type equations
7
therefore, for (t, s) large enough, we have g(t, s) → −∞. So there is a pair of (tu , su ) such that
g(tu , su ) = max g(t, s).
t,s≥0
We next claim that tu , su > 0. Indeed, without loss of generality, assuming the pair of (tu , 0) is
a maximum point of g(t, s), we get that
∂
λ
g(tu , s) = asku− k2 + hκ (tu u+ + su− )(4t2u sku+ kku− k2 + 4s3 ku− k2 )
∂s
4
Z
λs 0
+ 2 hκ (tu u+ + su− )ktu u+ + su− k4 ku− k2 −
f (su− )u− dx
N
2κ
R
Z
λs
− 2
− −
+
− 4
≥ asku k −
f (su )u dx − 2 ktu u + su k ku− k2 ,
N
κ
R
(2.5)
∂
since condition ( f 2 ), for λ, s enough small, we see that ∂s
g(tu , s) > 0, which implies that
g(tu , s) is increasing for s small. This contradicts that the pair of (tu , 0) is a maximum point of
g(t, s). Consequently, (tu , su ) is a positive maximum point of g(t, s).
Finally, we prove that tu u+ + su u− ∈ Mλ,κ . According to the definition of Φ, we note that
+
tu u + su u− ∈ Mλ,κ is equivalent to Φ(t, s) = 0 for any t, s > 0. Because the pair of (tu , su ) is
a positive maximum point of g(t, s), we observe that
∂
∂
g(t, s)|(tu ,su ) = g(t, s)|(tu ,su ) = 0,
∂t
∂s
is equal to
0
0
(tu u+ + su u− ), u+ i = h Jλ,κ
(tu u+ + su u− ), u− i = 0,
h Jλ,κ
which is same as
Φ(tu , su ) = 0.
Thence, by virtue of the definition of nodal Nehari manifolds, we show that tu u+ + su u− ∈
Mλ,κ , which finishes the proof.
Corollary 2.4. For any u ∈ H \{0}, then there exists a tu ∈ R+ such that tu u ∈ Nλ,κ for λ small. In
particular, Nλ,κ 6= ∅ and for all t ∈ R+ , there is
Jλ,κ (tu u) ≥ Jλ,κ (tu).
Lemma 2.5 (see Lions [14, 15]). Let r > 0 and p ∈ [2, 2∗ ). If {un } is bounded in H and
lim sup
n → ∞ y ∈R N
Z
Br (y)
|un | p dx = 0,
then we have un → 0 in Lq (R N ) for q ∈ (2, 2∗ ).
Lemma 2.6. Let {un } ⊂ Nλ,κ be a minimum sequence of Jλ,κ at level ceλ,κ , then {un } is bounded in H.
Proof. Arguing by contradiction, suppose kun k → ∞ as n → ∞, and set vn := kuunn k . Then there
exists a v ∈ H such that vn * v in H, up to a subsequence. Moreover, for p ∈ [2, 2∗ ), we have
either {vn } is vanishing, i.e.,
Z
lim sup
n → ∞ y ∈R N
Br (y)
|vn | p dx = 0
X. Yao and C. Mu
8
or non-vanishing, i.e., there exist r, δ > 0 and a sequence {yn } ⊂ R N such that
lim
Z
n→∞ Br (yn )
|vn | p dx ≥ δ > 0.
We next shall prove neither vanishing nor non-vanishing occurs and this will provide the
desired contradiction. If {vn } is vanishing, by Lemma 2.5, this implies vn → 0 in Lq (R N ) for
q ∈ (2, 2∗ ). Then, for every t > 0, we have, in view of ( f 1 ), ( f 2 ) and Sobolev’s inequality,
ceλ,κ + o (1) = Jλ,κ (un ) ≥ Jλ,κ (tvn )
at2
λ
kvn k2 + h(vn )kvn k4 −
F (tvn )dx
2
4
RN
Z
Z
at2
− εt2
v2n dx − Cε tq
|vn |q dx
≥
N
N
2
R
R
Z
at2
− εC1 t2 − Cε tq
|vn |q dx
≥
2
RN
at2
→
− εC1 t2 ,
2
Z
=
as n → ∞. This yields a contradiction for enough large t .
Should non-vanishing occur, we then check that for enough large n, by ( f 3 )
Jλ,κ (un )
a
F (un )
= −
|vn |2 dx
0≤
2
N
kun k
2
u2n
R
Z
Z
F (un )
F (un )
a
2
dx
−
|
v
|
|vn |2 dx
≤ −
n
2
2
2
|un |≤ M un
|un |> M un
Z
a
≤ −M
|vn |2 dx
2
|un |> M
Z
a
≤ −M
|vn |2 dx
2
[|un |> M]∩ Br (yn )
Z
a
≤ −M
|vn |2 dx
2
Br (yn )
a
≤ − Mδ
2
< 0,
Z
where M is enough large. This is a contradiction and completes the proof.
Lemma 2.7. Let {un } ⊂ Mλ,κ be a minimum sequence for Jλ,κ at level cλ,κ , then {un } has a convergent subsequence in H.
Proof. Let {un } ⊂ Mλ,κ be such that
Jλ,κ (un ) → cλ,κ ,
as n → ∞.
Then, by Lemma 2.6, we know that un is bounded in H and there exists a u ∈ H, up to a
subsequence, such that
un * u
in H,
un → u
in L p (R N ) for p ∈ (2, 2∗ ),
un → u
a.e. in R N .
(2.6)
Existence of sign-changing solution of Kirchhoff-type equations
9
From ( f 1 ) and ( f 2 ), we have, for ε small,
| f (t)| ≤ ε|t| + Cε |t|q−1 ,
for any t ∈ R,
(2.7)
thus by Hölder’s inequality and Sobolev’s inequality, we get
Z
Z
f (un )(un − u)dx ≤ ε|un |2 |un − u|2 + Cε
RN
≤
|un |q−1 |un − u|dx
RN
q −1
ε|un |2 |un − u|2 + Cε |un |q |un − u|q .
(2.8)
Thus thanks to boundedness of {un } in H and (2.6), we obtain that
Z
RN
f (un )(un − u)dx → 0
as n → ∞.
Then note that for n enough large,
0
o (1) = h Jλ,κ
(un ), un − ui = a(un , un − u) + λhκ (un )kun k2 (un , un − u)
Z
λ 0
4
+ 2 hκ (un )kun k (un , un − u) −
f (un )(un − u)dx
(2.9)
2κ
RN
λ
= a + λhκ (un )kun k2 + 2 hκ0 (un )kun k4 (un , un − u) + o (1).
2κ
It forces, as n → ∞,
λ 0
4
2
a + λhκ (u)kun k + 2 hκ (un )kun k (un , un − u) = o (1).
2κ
From the definition of h, we easily obtain (un , un − u) → 0 and kun k → kuk. Combining this
with (2.6), we demonstrate that un → u in H. This finishes the proof.
When {un } ⊂ Nλ,κ , using similar procedure of the proof above, we know that result of
Lemma 2.7 also holds at level ceλ,κ .
Lemma 2.8. The cλ,κ is attained by some u ∈ Mλ,κ for λ small, which is a critical point of Jλ,κ in H.
Proof. Let {un } ⊂ Mλ,κ be such that Jλ,κ (un ) → cλ,κ as n → ∞. By Lemma 2.7, we know that
there exists a u ∈ H such that
un → u,
u+
n → v,
u−
n
(2.10)
→ w,
in H as n → ∞. Since un ∈ Mλ,κ ,
2
aku+
nk
+ λhκ (un )kun k
2
2
ku+
nk
λ
+ 2 hκ0 (un )kun k4 ku2n =
2κ
Z
RN
+
f (u+
n ) un dx
Then, by ( f 1 ), ( f 2 ) and Sobolev’s inequality, we have
2
4
+ 2
aku+
n k − 4λκ ≤ a k un k −
Z
Z
λ
2
2
q
k u n k4 k u +
|u+
|u+
nk ≤ ε
n | dx + Cε
n | dx
N
N
κ2
R
R
2
+ q
≤ εC1 ku+
k
+
C
k
u
k
.
2
n
n
(2.11)
X. Yao and C. Mu
10
−
So ku+
n k ≥ C3 > 0, similarly, k un k ≥ C4 > 0. This implies that v, w 6 = 0. Since H is a Hilbert
space and the project mapping u 7→ u± is continuous in H, we get u+ = v and u− = w, then
u = u+ + u− is a sign-changing function. Next we prove u ∈ Mλ,κ . From un ∈ Mλ,κ , note
that
0
0
−
h Jλ,κ
( u n ), u +
n i = h Jλ,κ ( un ), un i = 0,
by (2.10) and passing to the limit, we obtain
0
0
h Jλ,κ
(u), u+ i = h Jλ,κ
(u), u− i = 0,
which implies u ∈ Mλ,κ and Jλ,κ (u) = cλ,κ . Consequently, Jλ,κ |Mλ,κ attains its minimum at u,
then u is a nontrivial critical point of Jλ,κ in Mλ,κ .
It remains to see that u is a critical point of Jλ,κ in H. Because u is a critical point of Jλ,κ in
0 ( u ) = 0 in M . Moreover, there exists a Lagrange multiplier µ such
Mλ,κ , we have that Jλ,κ
λ,κ
that
0
Jλ,κ
(u) − µΨ0 (u) = 0,
(2.12)
0 ( u ), u i. It suffices to prove that µ = 0. By (2.12), we have
where Ψ(u) = h Jλ,κ
0
h Jλ,κ
(u), vi − µhΨ0 (u), vi = 0,
for any v ∈ H.
(2.13)
Taking v = u, we compute that
λ
5λ
f 0 (u)u2 + f (u)udx
hΨ0 (u), ui = 2akuk2 + 4λhκ (u)kuk4 + 2 hκ0 (u)kuk6 + 4 hκ00 (u)kuk8 −
κ
κ
RN
Z
4 0
1 00
4
6
8
= λ 2hκ (u)kuk + 2 hκ (u)kuk + 4 hκ (u)kuk −
f 0 (u)u2 − f (u)udx
κ
κ
RN
Z
f 0 (u)u2 − f (u)udx.
≤ λ 8κ 4 + 64κ 4 + 16κ 4 hκ00 (u) −
Z
RN
In virtue of ( f 4 ), we know that there exists a positive constant α such that
Z
RN
f 0 (u)u2 − f (u)udx ≥ α > 0.
Therefore, hΨ0 (u), ui < 0 for enough small λ, together with (2.13), it showes that µ = 0. The
proof is completed.
Corollary 2.9. The ceλ,κ is attained by some u ∈ Nλ,κ , which is a critical point of Jλ,κ in H.
The proof is similar to that of Lemma 2.8, hence it is omitted here.
3
Proof of main results
According to the lemmas and corollaries in Section 2, we easily obtain the following results.
Theorem 3.1. Assume the conditions ( f 1 )–( f 4 ) hold, for λ small, functional Jλ,κ possesses one least
energy critical point uλ which is constant sign and one least energy sign-changing critical point vλ .
Moreover, the energy of the sign-changing critical point is strictly greater than the least energy, that is,
cλ,κ = Jλ,κ (vλ ) > Jλ,κ (uλ ) = ceλ,κ > 0.
Existence of sign-changing solution of Kirchhoff-type equations
11
Proof. By the the lemmas and corollaries in Section 2, we know that Jλ,κ possesses a least
energy critical point uλ and a least energy sign-changing critical point vλ .
+
For v+
λ , in view of the foregoing discussions, there exists a t = t ( vλ ) > 0 such that
+
tvλ ∈ Nλ,κ , then
+
−
+
−
0 < ceλ,κ = Jλ,κ (uλ ) ≤ Jλ,κ (tv+
λ ) = Jλ,κ ( tvλ + 0vλ ) < Jλ,κ ( vλ + vλ ) = cλ,κ .
Finally, we will prove that uλ is constant sign. Suppose that uλ is sign-changing, then uλ ∈
Mλ,κ and
ceλ,κ = Jλ,κ (uλ ) ≥ Jλ,κ (vλ ) = cλ,κ > ceλ,κ ,
this is absurd. We complete the proof.
Next we give an important identity to obtain that uλ and vλ are bounded uniformly in H.
That is a Pohožaev type identity, which was proved in [11, Lemma 2.6], here we omit the
details.
Lemma 3.2. If u ∈ H is a weak solution of
λ 0
2
4
a + λhκ (u)kuk + 2 hκ (u)kuk [−∆u + bu] = f (u),
2κ
x ∈ RN ,
(3.1)
then for λ small, the following Pohožaev type identity holds
Z
Z
Nb
λ 0
N−2
4
2
2
2
|∇u| dx +
|u| dx
a + λhκ (u)kuk + 2 hκ (u)kuk
2
2 RN
2κ
RN
=N
Z
RN
F (u)dx. (3.2)
Lemma 3.3. For uλ and vλ obtained in Theorem 3.1, if κ > 0 is large enough and λ > 0 is sufficiently
small, then uλ and vλ are bounded in H, that is, kuλ k, kvλ k ≤ κ.
Proof. This result was proved in [11, Lemma 2.7]. However, it plays a key role in proving
Theorem 1.1 and for the sake of completeness and convenience to reader, we here give the
detail. From Jλ,κ (vλ ) = cλ,κ , we also write it as
1
1
aN kvλ k2 + Nhκ (vλ )kvλ k4 − N
2
4
Z
RN
F (vλ )dx = cλ,κ N
(3.3)
0 ( v ) = 0, we know that (3.2) holds. Combining (3.2) and (3.3), we get that, for λ small,
By Jλ,κ
λ
Z
Z
a
λ 0
2
4
2
|∇vλ | dx ≤ a + λhκ (vλ )kvλ k + 2 hκ (vλ )kvλ k
|∇vλ |2 dx
2 RN
2κ
RN
(3.4)
λN 0
λ
4
6
= cλ,κ N + Nhκ (vλ )kvλ k + 2 hκ (vλ )kvλ k .
4
4κ
Now we start to estimate the right hand side of (3.4). As the procedure in the proof of
Lemma 2.3, we have, by the definition of h,
cλ,κ ≤ Jλ,κ ( ϕ + ψ)
Z
a
a
λ
= k ϕk2 + kψk2 + hκ ( ϕ + ψ)k ϕ + ψk4 −
F ( ϕ + ψ)dx
2
2
4
RN
Z
a a
λ
= + −
F ( ϕ + ψ)dx + hκ ( ϕ + ψ)k ϕ + ψk4
2 2
4Z
RN
Z
a a
≤ + − C1
ϕ2 dx − C1
ψ2 dx + C + λκ 4
2 2
BR (0)
BR (0)
≤ C1 + λκ 4 .
X. Yao and C. Mu
12
We also have that
λ
Nhκ (vλ )kvλ k4 ≤ λNκ 4 ,
4
and
λN 0
h
(
v
)
kvλ k6 ≤ 4λNκ 4 .
λ
κ
2
4κ
Then together with (3.4), we have
a
2
Z
RN
|∇vλ |2 dx ≤ NC2 + 6λNκ 4 .
0 ( v ) = 0, we have
Since Jλ,κ
λ
akvλ k2 + λhκ (vλ )kvλ k4 +
λN 0
h (vλ )kvλ k6
4κ 2 κ
Z
=
RN
f (vλ )vλ dx ≤ ε
Z
RN
v2λ dx
+ Cε
Z
∗
RN
v2λ dx. (3.5)
∗
Therefore, by D 1,2 (R N ) ⊂ L2 (R N ) and Sobolev’s inequality,
( a − ε)kvλ k2 ≤ Cε
≤ C3
Z
∗
ZR
N
RN
v2λ dx −
λN 0
h (vλ )kvλ k6
4κ 2 κ
|∇vλ |2 dx + 8λκ 4
≤ C4 ( NC2 + 6λκ 4 )2
∗ /2
(3.6)
+ 8λκ 4 .
Arguing by contradiction, suppose kvλ k ≥ κ. Then, by (3.6), we have
κ 2 ≤ kvλ k2 ≤ C5 ( NC2 + 6λκ 4 )2
∗ /2
+ 8C6 λκ 4 ,
which is impossible with κ large and λ small. So kvλ k ≤ κ, similarly, we get kuλ k ≤ κ. The
proof is finished.
In what follows, we start to prove Theorem 1.1.
Proof of Theorem 1.1. Let κ and λ be large and small, respectively. By Theorem 3.1, we know
that Jλ,κ possesses a least energy critical point uλ at level ceλ,κ and a least energy sigh-changing
critical point vλ at level cλ,κ , and according to Lemma 3.3, we obtain kuλ k, kvλ k ≤ κ, then
Jλ,κ = Jλ and uλ ,vλ are critical point critical of Jλ at level ceλ and cλ , respectively. Therefore, equation (1.1) has a least energy signed solution uλ and a least energy sigh-changing
solution vλ .
Finally, we will see the energy of sign-changing solution is strictly more than the least
energy. From Jλ,κ = Jλ and Theorem 3.1, we have
cλ = Jλ (vλ ) > Jλ (uλ ) = ceλ > 0.
Thus the proof is complete.
Existence of sign-changing solution of Kirchhoff-type equations
13
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